The basis for all the common methods for the analysis of passive calorimetric probe measurements is the consideration of temporal temperature changes during the heating and the cooling phase of the calorimetric probe [9]:

$$ {C}_p\frac{\mathrm{d}{T}_p}{\mathrm{d}t}={P}_{in}-{P}_{out} $$

(1)

with the heat capacity of the probe *C*
_{
p
}, the time derivative of the probe temperature *T*
_{
p
}. *P*
_{
in
} is the net incoming power containing the heating processes as well as losses due to plasma-wall-interactions. A list of the various contributions to the thermal energy balance at the substrate (probe) can be found in [12, 13]. *P*
_{
out
}, on the other hand, includes all loss processes dependent on the rising temperature of the probe. The processes are heat radiation (Stefan-Boltzmann law)

$$ {P}_{rad}=\varepsilon \sigma {A}_p\left({T}_p^4-{T}_{eq}^4\right), $$

(2)

with the emissivity *ε*, the Stefan-Boltzmann constant *σ*, the area of the probe *A*
_{
p
} and the equilibrium temperature *T*
_{
eq
}, and thermal conduction

$$ {P}_{cond}=c\left({T}_p-{T}_{eq}\right), $$

(3)

with a constant *c* depending on the material and geometry of the probe.

It includes, furthermore, the virtual reduction of the incoming power due to higher substrate temperatures. This reduction is often considered to be a loss in order to distinguish this effect from any other changes of the energy influx [4, 10]. As soon as the substrate temperature rises, the contributions of the total energy flux to the substrate which are dependent on the substrate temperature decline, generally. For example, under atmospheric pressure one of these contributions is that of the heated neutral gas. The energy transferred from the gas to the substrate will diminish with a rising substrate temperature.

At low pressure free convection can be neglected. When using atmospheric pressure plasma jets, forced convection, which can be treated linearly analog to thermal conduction, may become dominant during the plasma treatment.

The three loss processes, radiation, conduction, and convection, are also those which are present in the cooling phase after plasma treatment. Here, the convection losses are dependent on the experimental conditions. If the gas flow is switched off together with the plasma or if the plasma jet is moved away from the probe, free convection is present. If the gas flow is still on, the losses caused by forced convection depend on the temperature of the gas.

The incoming power can be determined by a comparison of the change of temperature during the heating (*P*
_{
in
} – *P*
_{
out
}, plasma/energy influx on, eq. 4.1) and the cooling phase (*P′*
_{
out
}, plasma/energy influx off, eq. 4.2) leading to eq. 4.3:

$$ {C}_p{\left(\frac{\mathrm{d}{T}_p}{\mathrm{d}t}\right)}_{heat}={P}_{in}-{P}_{out} $$

(4.1)

$$ {C}_p{\left(\frac{\mathrm{d}{T}_p}{\mathrm{d}t}\right)}_{cool}=-{P}_{out}^{\prime } $$

(4.2)

$$ {P}_{in}={C}_p\left({\left(\frac{\mathrm{d}{T}_p}{\mathrm{d}t}\right)}_{heat}-{\left(\frac{\mathrm{d}{T}_p}{\mathrm{d}t}\right)}_{cool}\right)+\left({P}_{out}-{P}_{out}^{\prime}\right) $$

(4.3)

It is important to point out that the power loss functions during the two phases *P*
_{
out
} (*T*
_{
p
}) and *P′*
_{
out
} (*T*
_{
p
}) are not necessarily the same. A crucial point for the losses to remain the same is the reduction of the incoming power due to higher substrate temperatures. In the case of a plasma jet this loss is mainly caused by the temperature difference between the probe and the neutral gas. As soon as the probe temperature rises, the difference becomes smaller and the incoming power is reduced. To mirror this effect during the cooling phase, a gas flow equally large to that one during the heating phase and at equilibrium temperature has to be applied. However, this is often not achievable due to residual heat in the jet or other effects on the gas temperature. Turning the gas flow off leads to free convection and, thereby, again to different power loss contributions.

Another issue is the temperature of the surrounding, i.e., the probe holder. The power losses in eqs. 2 and 3 are dependent on this temperature, which is the equilibrium temperature *T*
_{
eq
} at the beginning of the treatment. If the probe holder heats up during the measurement these losses are no longer solely dependent on the probe temperature *T*
_{
p
}. Hence, a time dependent component is introduced and, therefore, the power loss functions *P*
_{
out
} (*T*
_{
p
}) and *P′*
_{
out
} (*T*
_{
p
}) differ from each another.

To illustrate these relations and the consequences for the measurement of high energy influxes, we start with the methods used for the analysis of small energy influxes and discuss their applicability to high energy influxes.

The simplest approach to apply the relation given in eq. 4.3 is the linear method. At the beginning of the heating curve at the equilibrium temperature *T*
_{
eq
} the losses are negligible and the energy influx is proportional to the slope of the temperature curve leading to [9]:

$$ {P}_{in}={C}_p{\left.{\left(\frac{\mathrm{d}{T}_p}{\mathrm{d}t}\right)}_{heat}\ \right|}_{T={T}_{eq}} $$

(5)

Since this method is applied at the beginning of the heating curve, the incoming power should be fully applied in a short time and the recording frequency of the probe temperature should be sufficiently high to minimize uncertainties for the determination of the slope \( {\left(\frac{\mathrm{d}{T}_p}{\mathrm{d}t}\right)}_{heat} \). For higher energy influxes the time, where the probe temperature is close to the equilibrium temperature, becomes very short and it is not always possible to apply the full power to the probe in the time available. This may be attributed to a starting phase of the source plasma or moving it over the probe if it was started elsewhere to account for the starting phase. There will always be a short time where the probe is exposed only to a fraction of the total energy flux of the source and where it is already heated up.

Another method is the exponential analysis. With the assumptions of a constant incoming power, linear loss terms, and a constant temperature for the surrounding of the probe the shape of the signal for heating and cooling phases will be exponential [9, 10]:

$$ {T}_{p, heat}(t)=\left({T}_{eq}+\frac{P_{in}}{a}\right)-\left(\frac{P_{in}}{a}\right) \exp \left(-\frac{a}{C_p}t\right) $$

(6.1)

$$ {T}_{p, cool}(t)={T}_{eq}+\left({T}_{p,st}-{T}_{eq}\right) \exp \left(-\frac{a}{C_p}t\right) $$

(6.2)

Here *a* is the constant for the linear cooling processes and *T*
_{
p,st
} the temperature of the probe at the beginning of the cooling phase.

Because of the constant temperature of the surroundings, i.e., *T*
_{
eq
}, the power loss functions *P*
_{
out
} (*T*
_{
p
}) and *P′*
_{
out
} (*T*
_{
p
}) are equal and cancel each other out (eq. 4.3) leading to [4]:

$$ {P}_{in}={C}_p\left({\left(\frac{\mathrm{d}{T}_p}{\mathrm{d}t}\right)}_{heat}-{\left(\frac{\mathrm{d}{T}_p}{\mathrm{d}t}\right)}_{cool}\right) $$

(7)

In this case, the incoming power can be simply calculated with the difference of the time derivatives of the temperature \( {\overset{.}{T}}_p=\frac{\mathrm{d}{T}_p}{\mathrm{d}t} \) during the heating and the cooling phase (Fig. 1a). Fig. 1b shows this relation for an exemplary measurement in a low pressure calibration experiment with an electron beam. It can be seen, that the curves of \( {\overset{.}{T}}_p \) for the heating and cooling phase are linear and parallel, so that their difference is roughly constant.

The assumptions made in this case are valid for short measurements with small energy influxes where the temperature of the surrounding materials, i.e., the probe holder, does not change significantly and for probe temperatures where heat loss by radiation is negligible.

If the probe is used for measurements at higher temperatures, i. e. heated up by sources with high energy influx, thermal radiation becomes relevant and the energy losses can no longer be assumed to be linear (see eq. 2). But, if the energy losses are solely dependent on the probe temperature (constant *T*
_{
eq
}), the function of the power losses *P*
_{
out
} represented by the cooling curve \( {\overset{.}{T}}_p \) is still the same for each energy influx measurement. Therefore, eq. 7 can be applied [12].

However, if plasma sources with a high energy influx onto substrates are used, the boundary conditions change. The assumption inherent to the cases above that the power losses are only dependent on the probe temperature can only be made if the surroundings of the probe, i.e., the probe holder, are at equilibrium temperature *T*
_{
eq
}. But due to the high energies the temperature of the holder may increase remarkably during the treatment. Hence, the temperature *T*
_{
eq
} in eqs. 2 and 3 is time dependent and, thereby, the power loss function *P*
_{
out
}. This means that the power loss functions *P*
_{
out
} (*T*
_{
p
}) and *P′*
_{
out
} (*T*
_{
p
}) in eqs. 4.1 and 4.2 are not the same. As a consequence, eq. 7 can no longer be used for the whole measurement and the method has to be modified.