# Turning Pan-Sensor Contact Variability into a Calibration Tool for Sub-1% Accuracy in Heat Capacity Measurements with Disk-Type DSC

Yahya Rharbi, Frédéric Hugenell

► To cite this version:

Yahya Rharbi, Frédéric Hugenell. Turning Pan-Sensor Contact Variability into a Calibration Tool for Sub-1% Accuracy in Heat Capacity Measurements with Disk-Type DSC. Thermochimica Acta, 2025, 755, pp.180192. 10.1016/j.tca.2025.180192. hal-05301616v2

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https://cnrs.hal.science/hal-05301616v2

Submitted on 29 Nov 2025

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# Turning pan-sensor contact variability into a calibration tool for Sub-1% accuracy in heat capacity measurements with disk-type DSC

Yahya Rharbi* ID, Frédéric Hugenell

Univ. Grenoble Alpes, CNRS, Grenoble INP, LRP, 38000 Grenoble, France

## ARTICLE INFO

### Keywords:

- Specific heat capacity

- Heat flux DSC

- Thermal contact resistance

- Step-scan DSC

- Uncertainty reduction

- Calibration

## ABSTRACT

The accuracy of absolute heat capacity ($c_p$) measurements in disk-type differential scanning calorimetry (DSC) has been perennially compromised by a fundamental yet unresolved issue: the poor reproducibility of pan-sensor thermal contact resistance (PS-TCR) upon manual repositioning. This systematic error, frequently treated as random noise, routinely introduces significant uncertainties that limit the reliability of $c_p$ data, especially for small samples.

This work introduces a methodological advancement to overcome this limitation. We demonstrate that the isothermal heat flow ($Q_{iso}$), a readily measurable parameter in step-scan protocols, serves as a highly sensitive and quantitative signature of pan-positioning artifacts. By establishing a robust linear correlation between $Q_{iso}$ and the integrated enthalpy signal ($Q_s$), we develop a novel correction procedure that effectively isolates and subtracts the positioning-dependent error component.

This strategy, validated experimentally and supported by a refined thermal model, systematically reduces $c_p$ measurement uncertainty to below 1 % for sample masses as low as 5 mg, a level of precision previously unattainable in conventional disk-type DSCs without specialized equipment. Crucially, this approach requires no hardware modifications, relying solely on software-based post-processing of existing step-scan data. Our findings provide a practical and universally applicable solution to a decades-old problem, finally enabling high-accuracy $c_p$ determination on standard DSC instruments and democratizing precision for the broader materials science community.

## 1. Introduction

Apart from the most reliable methods for specific heat capacity ($c_p$) determination, such as adiabatic calorimetry, Differential Scanning Calorimetry (DSC) is widely regarded as a practical and accessible alternative due to its availability and versatility [1,2]. Nevertheless, achieving $c_p$ measurement accuracy below 1 % by DSC remains a persistent challenge [1,3–10]. Considering the intrinsic performance of modern DSC instruments, such precision should, in principle, be attainable for a typical user [6,11,12]. In practice, however, this level of accuracy is not consistently achieved, especially with disk-type heat-flux DSCs, and becomes particularly challenging when measuring low-signal materials, such as small sample masses [1,3,13–20]. The precision of heat capacity ($c_p$) measurements in disk-type DSCs is notably variable, with literature reporting uncertainties typically amounting to few percent and not uncommonly surpassing 10 % [1,13,16,17,21]. A primary contributor to this variability is the poor reproducibility of

pan-sensor thermal contact upon pan repositioning, which can cause uncertainties to increase significantly [13,15,16]. Although widely recognized, this challenge has not been fully resolved. Consequently, measurement precision remains critically dependent on operator skill and consistency, which fundamentally limits the reliability and universal accessibility of high-accuracy $c_p$ data.

The precision of modern DSC instruments is theoretically sufficient for accurate $c_p$ determination; however, the requisite three-step measurement protocol—involving repeated furnace openings for baseline, sample, and standard measurements—introduces the dominant source of error [22]. Each handling step perturbs the thermal equilibrium of the assembly, but the most significant effect arises from the inability to perfectly reproduce the pan-sensor contact upon repositioning. This is conclusively demonstrated by the disparity between machine repeatability (errors < 0.2 % without intervention) and experimental reproducibility (variations of several percent, including pan handling) [13, 22]. This orders-of-magnitude difference confirms that pan-sensor

<footnote>* Corresponding author.</footnote>

<footnote>E-mail address: Yahya.Rharbi@univ-grenoble-alpes.fr (Y. Rharbi).</footnote>
contact variability is the primary external limitation, creating a critical need for strategies to control it and thereby reduce a major component of the $c_p$ uncertainty budget [23].

The central challenge appears to be associated with the pan-sensor thermal contact resistance (PS-TCR), a parameter likely influenced by surface asperities and roughness, contact pressure, trapped air gaps, and other geometric heterogeneities at the interface [24–26]. Although DSC manufacturers have designed pans and sensors to optimize contact reliability, the PS-TCR remains difficult to standardize. In disk-type heat-flux DSCs, the pan is manually centered on the sensor, introducing an inherent and variable PS-TCR. Unlike other thermal resistances in DSC, which are fixed by design or can be actively controlled (e.g., via gas atmosphere), the PS-TCR is predominantly sensitive to pan-sensor surface topography and placement accuracy. Minute deviations can introduce thermal asymmetry, affecting the balance between the measuring and reference sides and propagating error into subsequent data analysis. Manufacturers have developed locating pins and ergonomic pans to improve positioning reproducibility, but mechanical clearance remains necessary, preventing perfect repeatability. Furthermore, despite their availability, these solutions are not yet routinely used in common practice, as evidenced by a bibliographic survey showing that over 80 % of published DSC studies still rely on standard flat-bottom pans, thereby embedding positioning-related errors into reference data. The availability of robotic autosamplers, which could further mitigate these effects, also remains limited, leaving reproducibility largely dependent on operator skill and consistency.

Strategies to improve zeroline subtraction were developed, including additional temperature sensors and algorithmic corrections for baseline drift [27–29]. While useful, these methods do not address the core issue of operator-induced PS-TCR variability during pan repositioning, as reproducibility issues persist even with such modifications [13]. Their effectiveness relies on accurate thermal parameters and typically assumes thermal asymmetries remain constant - an invalid assumption when manual handling alters the PS-TCR. Thus, residual errors persist in the corrected $c_p$ data when pan-sensor contact is inconsistent.

Thus, pan repositioning-induced PS-TCR variation continues to fundamentally limit $c_p$ measurement accuracy in disk-type DSCs. Despite design improvements, this persistent effect [13] is often misclassified as random error rather than the systematic uncertainty it represents [15]. Moving beyond its recognition to actionable solutions requires a systematic quantification of PS-TCR variability under realistic conditions and its precise contribution to the overall $c_p$ uncertainty budget. This essential characterization will establish the foundation for developing effective mitigation strategies.

The present work systematically quantifies the influence of PS-TCR on the accuracy of $c_p$ measurement in disk type DSC. The variability introduced by manual pan positioning is quantified and its impact on heat-flow reproducibility analyzed, to establish a practical strategy for improving the precision of absolute heat capacity measurements using step-scan or constant heat rate protocols. Particular attention is given to the isothermal response and its temperature evolution, revealing a direct relationship between baseline fluctuations and the dynamic heat-flow signal. This enables development of a procedure that evaluates the error range associated with PS-TCR variability. Furthermore, the role of PS-TCR in facilitating residual heat exchange with the external environment is investigated, explaining the nonzero heat flow even under nominally isothermal conditions. Integrating these findings into a thermal model allows quantification of both PS-TCR variability and external heat leakage, clarifying their combined impact on the accuracy of $c_p$ determinations.

Beyond quantification, we propose a practical calibration procedure that explicitly accounts for PS-TCR variability and mitigates operator-induced errors. This approach systematically reduces $c_p$ measurement uncertainty to within 1 % or better, even for sample masses typically challenging for disc type DSC. Crucially, the strategy requires no

specialized accessories or robotic handling, enabling laboratories worldwide to significantly improve measurement precision using standard equipment. Ultimately, this makes high-accuracy absolute $c_p$ determination more accessible and reliable for the broader research community, particularly for small sample masses.

## 2. Experimental

### 2.1. Materials

The materials used in this study include polystyrene (Ps, Chemical Pressure Mw 207.7 kg/mol, Mw/Mn = 1.05), Laropal A81 (LA81 supplied by BASF), Aluminium (ships, 99.99 %, Mettler Toledo) and sapphire (Mettler Toledo standard).

### 2.2. Differential scanning calorimetry (DSC) measurements

Thermal analysis was performed using a Mettler Toledo DSC 823e differential scanning calorimeter. The instrument was calibrated for temperature, heat flow, and thermal lag using high-purity indium and zinc standards. A constant purge gas flow of 60 ml/min (nitrogen or dry air) was maintained to ensure a stable atmosphere within the furnace.

Measurements were carried out using 40 µL aluminum pans. To systematically investigate the effect of pan positioning, both standard flat-bottom pans and self-centering ergofix pans were employed. The sample and reference pans were meticulously mass-matched, with a mass difference not exceeding 0.04 mg. Different configurations of 40 µL pans were used during this study: either ergofix and flat bottom pans both with and without lid. When not specified, the standard configuration used in the following is ergofix pan with lid. The mass range of this configuration is around 49 mg, and without lid yields around 32 mg.

The primary measurement protocol was a step-scan method, which consisted of a 3-minutes isothermal hold, followed by a temperature ramp of $\Delta T = 2$ K at a rate of 10 K/min, and a subsequent 3-minutes isothermal hold [5,30]. This sequence was repeated over the desired temperature range. While most experiments focused on the narrow interval of 0 °C to 2 °C to minimize temperature-dependent effects, additional measurements were conducted over broader ranges from -80 °C to 140 °C or from 0 °C to 140 °C. The isothermal heat flow value recorded at 0 °C at the beginning of each measurement, denoted $Q_{iso}$, served as a key reference metric for quantifying pan-placement artifacts. For comparison, some experiments were also performed under constant heating conditions at 1 K/min following an initial isothermal equilibration.

To assess reproducibility and isolate the error induced by manual handling, a specific repositioning procedure was implemented for each of the 3 steps previously mentioned (empty pan, sapphire and sample measurements). Each run began by opening the DSC furnace, removing the sample pan, and repositioning it to a new random position on the sample sensor. A new measurement was then initiated, always recording the initial isothermal heat flow at 0 °C. This procedure was repeated 10 times using the same sample. Crucially, the reference pan remained untouched throughout all 10 runs to ensure that any signal variability originated solely from the sample side.

For each of the ten experiments described above, the following data processing was performed. The recorded signals comprised the isothermal heat-flow levels at the end of the initial and final isotherms; the isothermal level $Q_{iso}$ of the initial isotherm was used to quantify pan placement. The transient heat-flow signal Q between the two isotherms was integrated over time to obtain the enthalpy signal $Q_s$. To isolate the material contribution, a linear baseline was constructed by interpolating between the two isothermal levels; this baseline heat flow was subtracted from Q prior to integration, and the baseline-corrected Q was integrated to yield $Q_s = \int \dot{Q}dt$.

This procedure yielded ten independent pairs of values ($Q_{iso}$, $Q_s$). The
variability in Q<sub>iso</sub> across the ten runs directly quantified the pan- placement error, while the variability in the final calculated specific heat capacity, derived from Q<sub>s</sub>, reflected the overall impact of this error on measurement reproducibility. In additional measurements, the same 10-run procedure with sample pan repositioning was repeated after systematically varying the position of the reference pan. This approach, performed for several distinct reference pan positions, allowed for the separate quantification of variability originating from both the sample and reference sides of the DSC assembly. Specific heat capacity is calculated using the classical method:

$$c_p = \frac{Q_s - Q_{s,0}}{Q_{s,saph} - Q_{s,0}} \cdot \frac{m_{saph}}{m_{\text{sample}}} \cdot c_{p,saph} \quad (1)$$

where c<sub>p,saph</sub> is the specific heat capacity of the sapphire standard, m<sub>saph</sub> is the sapphire mass, m<sub>sample</sub> is the sample mass, Q<sub>s</sub>, Q<sub>s,saph</sub> and Q<sub>s,0</sub> represent the enthalpy signals for the sample, sapphire standard, and empty pan, respectively. Two calculation approaches were employed. In the first approach, Eq. (1) was applied using single measurement values for Q<sub>s</sub>, Q<sub>s,saph</sub>, and Q<sub>s,0</sub> obtained from one complete run, thereby propagating all positioning-related uncertainties. In the second approach, the subtraction of Q<sub>s,0</sub> was performed while explicitly considering how sample pan positioning variability affects Q<sub>s</sub>, Q<sub>s,saph</sub> and Q<sub>s,0</sub>. This separation allows for a more detailed analysis of the individual contributions to measurement uncertainty.

# 3. Results and discussion

## 3.1. Signal uncertainty resulting from pan positioning

The first step in this analysis was to distinguish between machine repeatability and experiment reproducibility. Machine repeatability was assessed by repeating the identical measurement procedure under stable conditions, including a short isothermal hold, without sample removal. Measurement reproducibility, conversely, involved the complete experimental procedure, including furnace handling, sample placement, thermal equilibration, and the measurement itself. The repeatability of the enthalpy signal (Q<sub>s</sub>), assessed over a temperature range of 0 to 2 °C, was excellent. Ten consecutive measurements performed on a single sample over several hours showed negligible variation of Q<sub>s</sub>, with a standard deviation (δQ<sub>s</sub>) between 1 and 10 µJ. The lowest variation was observed for inert samples such as sapphire, while polymers (e.g., polystyrene or LA81) exhibited slightly higher variability. This is likely attributable to intrinsic thermophysical properties or subtle morphological changes in the polymers upon thermal cycling. The relative error for all these measurements remained below 0.3 % (Fig. 1), consistent with values reported for other DSC instruments using the same procedure [13]. This result confirms that the electrical stability of the DSC is much higher than the variability observed in repeatability tests, indicating that the dominant contribution to Q<sub>s</sub> variability arises from thermally induced effects associated with sample pan handling and pan-sensor contact conditions.

On the other hand, the reproducibility of Q<sub>s</sub>, assessed over ten independent experimental runs, showed significantly greater variability. The expanded uncertainty (with a coverage factor k = 2, representing a 95 % confidence level) reached a maximum of approximately 2 mJ when using self-centering "ergo" pans. However, this value increased to about 3 mJ when using standard flat bottom pans, even when users meticulously followed a strict protocol to center them as accurately as possible by eye. This procedure was tested with several trained users. Excessive decentering, which can occur in standard practice, led to even higher expanded uncertainty values, exceeding 3 mJ. Consequently, the analysis presented hereafter focuses solely on results obtained with ergofix pans, which rely on automatic centering to ensure consistency.

The relative error (δQ<sub>s</sub>/Q<sub>s</sub>) was calculated and is plotted in Fig. 1 as a function of the average Q<sub>s</sub>. For a Q<sub>s</sub> signal greater than 60 mJ

<figure><img src="image_3.png" /><figcaption>Fig. 1. Relative error of the measured enthalpy signal (Q<sub>s</sub>) as a function of the total enthalpy Q<sub>s</sub>, obtained from DSC measurements performed in step-scan mode, as described in the text. The measurements were carried out as a step-scan experiment from 0 °C to 2 °C at 10 K/min, with each temperature step preceded and followed by a 3 min isothermal segment. Errors are expressed as the relative standard deviation over 10 successive measurements conducted under different procedures. Most measurements were performed using 40 µL pans with ergofix; a few measurements were carried out with 40 µL flat-bottom pans, shown as dashed rectangles in the figure. The average pan mass was 49.5 mg with a mass difference of 0.04 mg between the sample and reference pans. Data were collected using different samples placed in pan 2: two sapphire disks, one sapphire disk, LA81 sample with varying mass (2.36 mg, 5.24 mg, 7.54 mg, and 15.33 mg), and polystyrene (PS) samples of 5.16 mg and 19.82 mg. PS measurements were performed using pans without ergo. (■ yellow) represent repeatability errors (δQ<sub>s</sub>/Q<sub>s</sub>) obtained without removing the sample pan. (◆ blue) represent reproducibility errors determined by removing and repositioning the sample pan between measurements. (■ orange) represent reproducibility errors calculated from heat capacity measurements, including the standard deviation of the empty pan and sapphire measurements. (▲ green,*) — represent errors on c<sub>p</sub> after correcting Q<sub>s</sub> by subtracting the empty-pan signal according to the procedure described in the text, values obtained from the empty-pan measurement, from the sample measurement, and from the empty-pan calculation based on Eq. (11). Dashed lines (—) represent the maximum repeatability error and (—) the maximum error after positional error subtraction.</figcaption></figure>

(corresponding to a 46 mg sapphire sample), the relative error was about 1.1 %, a value considered excellent for a disk-type DSC. However, as the sample mass or total heat capacity decreases (C<sub>p</sub>), the relative error increases progressively. It reaches approximately 2 % for a sapphire mass near 20 mg, ~ 3 % for polymer samples (PS or LA81) with a Q<sub>s</sub> of approximately 20 mJ, and exceeds 5 % for Q<sub>s</sub> values around 12 mJ (corresponding to a mass of ~ 5 mg of PS). This trend clearly defines a practical lower mass limit for reliable c<sub>p</sub> measurement with the current experimental setup.

As shown in Fig. 1, measurements performed with flat-bottom pans exhibit a slightly higher uncertainty, by a few percent, likely due to the increased variability in pan placement. When the pans are well aligned using the visual guide marks on the sensor, this variability remains moderate; however, less accurate centering can lead to a substantial increase, with cumulative errors exceeding 10 %, even for a sapphire standard, as reported [13]. Nevertheless, this centering procedure ultimately relies on human ability and remains inherently non-quantitative.

This reproducibility uncertainty is far greater than the potential error arising from mass differences between the sample and reference pans, a factor cited to justify the zero-line subtraction procedure. Indeed, these measurements were conducted using empty pans with a mass matched to within 0.04 mg of the sample pan, representing a mass error of <0.1 %. The systematic enthalpy offset introduced by this minimal mass difference, calculated as ΔQ<sub>0</sub> = c<sub>p</sub><sup>Alu</sup> Δm ΔT ≈ 8 µJ, is negligible when compared to the observed experimental variability of approximately 2 mJ through pan repositioning. Therefore, the difference in the heat capacity of the pans as a source of error can be safely neglected throughout
this analysis.

The impact of pan positioning on enthalpy-signal variability is equally critical for the empty-pan measurement used to define and subtract the baseline. Repeated empty-pan runs show that the spread in $Q_{s,0}$ is of the same order as, and can even exceed, the ten-run average $\langle Q_{s,0} \rangle$. For instance, even under optimal conditions using ergo pans (which limit positional variability) and mass-matched (within 0.04 mg), the observed range of $Q_{s,0}$ variability exceeds 2 mJ, while $\langle Q_{s,0} \rangle$ typically falls between -1 mJ and 1 mJ. This variability increases up to 3 mJ with standard flat pans, mirroring the behavior observed in sample measurements. Consequently, the relative error $\delta Q_{s,0}/\langle Q_{s,0} \rangle$ often surpasses 100 %, making a graphical representation of this ratio (e.g., in Fig. 1) non-meaningful.

Nevertheless, the variability of $Q_{s,0}$ is particularly critical for the error propagation in the specific heat capacity calculation based on Eq. (1). This is because, baseline subtraction in the conventional approach is performed without separately accounting for the roles of sample pan and empty pan placement variability during their respective measurements. This limitation motivated a rigorous uncertainty-propagation analysis (Eqs. (2)-(3)) to quantify and properly attribute these contributions. The combined uncertainty was evaluated by considering the independent uncertainties in the sample measurement ($\delta Q_s$), sapphire standard measurement ($\delta Q_{s,saph}$), and baseline measurement ($\delta Q_{s,0}$). The uncertainty associated with each difference term was calculated through quadrature summation:

$$ \delta(Q_s - Q_{s,0}) = \sqrt{(\delta Q_s)^2 + (\delta Q_{s,0})^2} \\ \delta(a(Q_{s,saph} - Q_{s,0}) = \sqrt{(\delta Q_{s,saph})^2 + (\delta Q_{s,0})^2} \quad (2) $$

The resulting relative uncertainty in $c_p$ is given by:

$$ e_r = \left( \frac{\delta(Q_s - Q_{s,0})}{Q_s - Q_{s,0}} \right)^2 + \left( \frac{\delta(Q_{s,saph} - Q_{s,0})}{Q_{s,saph} - Q_{s,0}} \right)^2 + \left( \frac{\delta m_{saph}}{m_{saph}} \right)^2 + \left( \frac{\delta m_{sample}}{m_{sample}} \right)^2 \\ \frac{\delta c_p}{c_p} = \sqrt{e_r} \quad (3) $$

The uncertainty in the calculated $c_p$ values, determined using Eqs. (2) and (3), is significantly affected by pan positioning variability inherent in the sample measurement ($\delta Q_s$) and is further amplified by the baseline subtraction procedure through ($\delta Q_{s,0}$). This amplification is visualized in Fig. 1, which compares the relative error calculated using non-baseline-corrected $\delta Q_s/Q_s$ with the baseline-corrected $\delta(Q_s - Q_{s,0})/(Q_s - Q_{s,0})$ derived from Eq. (2). Although $Q_{s,0}$ is subtracted from both numerator and denominator in Eq. (1), its associated uncertainty does not cancel out. Instead, the baseline measurement error contributes additively to both terms of the quotient, resulting in a synergistic amplification that combines sample positioning variability with baseline determination uncertainty.

Moreover, in the case of small sample masses, the conventional practice of baseline subtraction amplifies even more the total uncertainty since the average value $\langle Q_s \rangle$ decreases while $\delta(Q_s - Q_{s,0})$ is mass independent (Eq. (3)). These results explain the exaggerated errors observed in several literature reports of absolute $c_p$ measurements using disk-type DSC and make this method less recommended for the precise evaluation of this thermophysical property.

This analysis further reveals that the conventionally calculated error bars for $c_p$, derived from repeatability-based uncertainty propagation, significantly underestimate the true experimental uncertainty. When $c_p$ error is computed using standard repeatability data and cumulative error propagation, the resulting uncertainty appears deceptively low, below 0.5 % for most cases represented in Fig. 1. However, these values fail to capture the full extent of measurement error, as they do not account for systematic contributions arising from pan repositioning. The discrepancy between the calculated error and the actual reproducibility observed in practice underscores the limitations of relying solely on

repeatability-based uncertainty estimates in disk-type DSC measurements.

This error propagation process, inherent to the experimental reproducibility, remains consistent across all investigated temperatures, as demonstrated in Fig. 2 for polystyrene (5.16 mg) and a sapphire disk (20 mg) characterized using the step-scan protocol. The figure clearly shows the vertical shift in the enthalpy signal ($Q_s$) resulting from sample repositioning. These measurements were conducted using ergo-free pans where the possibility of excessive decentering is permitted. Even under conditions where pans were visually centered to the best of the operator's ability, the $Q_s$ signal exhibits vertical offset while maintaining relatively similar temperature-dependent shape. Only when the decentering becomes excessive and visually observable does the $Q_s$ versus temperature curve begin to show noticeable distortion in its line shape. This systematic behavior suggests that the signal variation due to different positioning can be considered predominantly additive, with an existing temperature-dependent evolution that demonstrates relatively

<figure><img src="image_4.png" /><figcaption>Fig. 2. Impact of pan positioning on enthalpy signal $Q_s$ measured using step-scan procedure. Data show $Q_s$ versus temperature for different sample pan repositioning sequences. Experiments employed sequential pan removal and repositioning at 0 °C before starting the DSC scan experiment. Experiment use standard flat-bottom pans without ergofix to maximize positioning variability, with sample and reference pans mass-matched within either 0.01 mg or 0.04 mg. Each measurement followed the sequence: 3 min isotherm, heating at 10 K/min for $\Delta T = 2$ K, 3 min isotherm at the new temperature, heating at 10 K/min for $\Delta T = 2$ K, and so on until reaching the final temperature. Temperature ranges spanned either 0-140 °C or -80-140 °C. Measurements were performed using: (a) polystyrene (PS, 5.16 mg); (b) sapphire (22.38 mg); (c) empty pans (48.69 mg and 48.70 mg). The observed variability in $Q_s$ with sample repositioning for the three samples demonstrates the significant impact of pan positioning on enthalpy measurement reproducibility.</figcaption></figure>
smaller amplitude compared to the error induced by vertical shifting. The additive nature of this positional error provides important insights for understanding the fundamental limitations of disk-type DSC measurements, particularly concerning the separation of instrumental artifacts from genuine material properties.

To further investigate pan positioning effects, complementary experiments were conducted using constant heating rate protocols (1 K/min) alongside the step-scan methodology. In these the heat flow signal Q was acquired vs. time during heating for different pan placements. The Q was normalized by subtracting the initial Qiso measured at 0 °C (Figure S2b). The steady-state heat-flow magnitude (Q - Qiso), evaluated 180-200 s after the ramp begins, depends strongly on pan placement. This positional dependence propagates to cP calculations via cP = (Q' - Qiso) / (β · msample), where β represents the heating rate. Even with Qiso variability constrained to below 100 µW for a 5.16 mg PS sample, the resulting cP uncertainty exceeds 10 %. Parallel measurements with empty pans confirm identical positional dependence (Figure S2a), demonstrating that pan positioning artifacts critically affect both step-scan and constant heating rate methodologies, with particularly pronounced impact in the latter case.

## 3.2. Role of pan position on the isotherm

Pan positioning systematically alters the isothermal heat-flow baseline, Qiso, measured before and after each temperature step in a step-scan DSC protocol. Each reseating of the sample pan (with the reference pan fixed) produces reproducible, position-dependent shifts in both Qiso and its temperature derivative biso = dQiso/dT (Fig. 3). Although Qiso exhibits some temperature dependence, this variation remains negligible within the small temperature increment (ΔT = 2 K) of each individual step. This pan-positioning effect occurs consistently across empty pans, sample-loaded pans, and reference materials, demonstrating its generalizable influence on thermal measurement. Crucially, these variations are both reproducible and position-specific; each distinct sample pan repositioning produces a unique yet stable deviation in Qiso, provided the reference pan configuration remains unchanged. This indicates that pan-handling artifacts constitute a structured error source manifesting as characteristic instrumental signatures. These findings establish Qiso as both a sensitive diagnostic tool for detecting positioning artifacts and a potential basis for correction strategies to improve accuracy in quantitative calorimetry using disk-type DSC systems.

The joint dependence of Qiso on temperature and pan placement was thoroughly quantified and modeled to further establish its utility as a positioning control parameter (Fig. 3). This analysis demonstrates that Qiso values monitored at the end of each isothermal segment exhibit significant variation in its temperature slope (below 0 °C) with pan position, confirming the parameter's sensitivity to mechanical placement. Moreover, all Qiso profiles converge toward a common point near -60 °C, indicating a fundamental thermal characteristic of the sensor assembly. Furthermore, above approximately 50 °C, the curves exhibit a distinct inflection whose direction, upward or downward, depends critically on the specific system configuration, particularly the centering of the sensor disc. To mathematically describe this behavior the temperature dependence of Qiso is characterized by a second-order polynomial: $Q'_{iso} = (T - T_f)(a(T - T_f) + b) + c$ where T<sub>f</sub> represents the crossover temperature (observed between -66 °C and -59 °C in our measurements). The linear coefficient b dominates the thermal response between -80 °C and room temperature, changing sign from negative to positive depending on the sample pan positioning.

These results provide further evidence that pan-positioning artifacts manifest as reproducible, mathematically quantifiable signatures in the isothermal signal, rather than random measurement noise. The consistent convergence behavior across all positions suggests an underlying physical mechanism related to the sensor's thermal field symmetry, while the positional dependence of the polynomial coefficients offers a

potential pathway for quantitative correction of these systematic errors.

The correlation between the enthalpy signal Qs and the isothermal heat flow Qiso is clearly demonstrated in Fig. 4, which plots Qs-(Qs) against Qiso for a Laropal sample measured via step-scan between 0 °C and 2 °C. Data were collected using a centered ergofix-type pan with controlled, limited displacement. Measurements include both the sample and an empty reference pan, with the sample pan randomly repositioned between runs to simulate typical reproducibility conditions. The results show that the relationship between Qs-(Qs) and Qiso follows the same trend for both the sample and the empty pan, indicating that the observed signal variation stems primarily from pan placement rather than sample-specific properties. This finding suggests that the contribution of pan placement can be mathematically isolated and removed from the measured Qs.

The slope of the Qs versus Qiso plot, defined as dQs/dQiso, ranges from approximately 15 s for empty pans to values above 17 s when sample

<figure><img src="image_5.png" /></figure>

**Fig. 3.** Impact of pan positioning on the isothermal heat flow Qiso during step-scan measurements. Data represent Qiso values recorded at different temperatures across multiple thermal steps following sample pan repositioning. The experimental protocol involved repositioning the sample pan at 0 °C prior to DSC measurement initiation. Measurements were carried out using standard flat-bottom pans without ergofix. Sample and reference pans were mass-matched within 0.01–0.04 mg. The thermal sequence consisted of: 3 min isotherm, heating at 10 K/min for ΔT = 2 K, 3 min isotherm, and repeating these steps across temperature ranges of 0–140 °C or -80–140 °C. Qiso is recorded at the end of each isotherm phase. Measurements include: (a) polystyrene (5.16 mg), (b) sapphire (22.38 mg), and (c) empty pans (48.69–48.70 mg). The observed Qiso variability with pan positioning establishes it as a reliable indicator of pan position. The temperature dependence of Qiso is fitted to polynomial functions of the form $Q'_{iso} = (T - T_f)[a(T - T_f) + b] + c$. where a, b, and c are fitting parameters; solid lines represent the best fit.
<figure><img src="image_6.png" /><figcaption>Fig. 4. Centered enthalpy signal $Q_s - (Q_s)$ plotted versus the isothermal heat-flow baseline $Q_{iso}$ for a sample pan repeatedly reseeded in the sensor (reference pan fixed). Measurements used ergo pans (Ergofix). Empty pans were mass-matched within 0.04 mg; the same pans were then loaded with LA81 (7.54 mg). Step-scan protocol between 0 °C and 2 °C, increments ΔT = 2 K executed at 10 K/min; the step was preceded and followed by a 3-min isothermal segment and $Q_{iso}$ is the value of the heat flow at the end of the initial isotherm. The solid line represents the best linear fit of $Q_s$ versus $Q_{iso}$ obtained from the simulation model, using the corresponding heat capacities of the empty pan and sample, the thermal resistances, the estimated $T_f$, and the leakage resistance $R_f$ as a fitting parameter (see in text). The simulations were performed using a thermal procedure similar to that of the experiment.</figcaption></figure>

(LA81 with total heat capacity $C_s \approx 0.01$ J/K) is added to the sample pan. These results are most reliable when $Q_{iso}$ is measured close to $T_f$, where its dependence on temperature remains linear.

Consistent trends are observed when $Q_s$ is plotted as a function of the slope of $Q_{iso}$ versus temperature ($b_{iso} = dQ_{iso}/dT$) evaluated below 0 °C (Figure S3). This approach yields comparable values of $dQ_s/db_{iso}$, approximately 900 s·K for empty pans and about 1000 s·K for pans containing LA81 with $C_s \approx 0.01$ J/K. Both $dQ_s/dQ_{iso}$ and $dQ_s/db_{iso}$ can therefore be used to quantify the dependence of $Q_s$ on pan positioning under conditions where $Q_{iso}$ remains linear with temperature. Using $b_{iso}$ provides improved robustness against temperature variations but requires longer measurement times due to the need for accurate determination of the $Q_{iso}$ temperature slope.

A key requirement for reliable measurements is to keep the reference pan fixed, especially in disk-type DSC where both pans share the same sensor that is highly sensitive to mechanical displacement. The effect of reference repositioning was evaluated by repeating the Fig. 4 protocol while systematically shifting the reference pan and then performing the sample-pan placement series. As shown in Figure S4, the calibration relation $Q_{s,0}$ versus $Q_{iso}$ retains the same slope but is shifted vertically with each reference position. This confirms that calibration must be performed strictly with an unmoved reference pan throughout the entire experiment; otherwise, perturbations of the reference introduce bias into the $Q_s$ vs $Q_{iso}$ calibration curve.

This indicates that the correction of $Q_s$ must account not only for the difference in total heat capacity due to the mass imbalance of the empty pans ($\Delta Q_0$) but also for the variability introduced by pan repositioning. In the present experiments, the contribution from $\Delta Q_0$ can be neglected, as it corresponds to only about 8 µJ. The observed variability in $Q_s$ can therefore be attributed predominantly to pan positioning. Considering the near-linear relationship between $Q_s$ and $Q_{iso}$, and the fact that the effect of pan positioning is embedded in the measurable parameter $Q_{iso}$, $Q_s$ can be corrected using a first-order linear approximation of the form:

$$Q_{s,corr} = Q_s - \frac{dQ_s}{dQ_{iso}} (\dot{Q}_{iso} - \dot{Q}_{iso,0}) \quad (4)$$

where $\dot{Q}_{iso}$ is the heat flow measured during the isothermal segment preceding the step scan for the sample measurement, and $\dot{Q}_{iso,0}$ is the extrapolated isothermal value corresponding to zero $Q_s$ for the empty pan. The slope $\frac{dQ_s}{dQ_{iso}}$ is taken from the empty-pan measurement. This correction significantly reduces the position-dependent bias in $Q_s$, as

evidenced by the marked decrease in the variability of $Q_{s,corr}$. The relative uncertainty, $\delta Q_{s,corr}/Q_s$, falls to approximately 1 % for enthalpy signals above 10 mJ, which corresponds to sample masses greater than ~5 mg for a typical polymer such as polystyrene (Fig. 1). This represents a marked improvement in precision for disk-type DSC systems, where uncertainties are typically much higher. Indeed, this level of reproducibility is comparable to that achieved with power-compensated DSC instruments using larger samples (50 mg) [12].

For very small samples (e.g., 2 mg of LA81, $Q_s < 10$ mJ), the corrected relative error remains around 3 %, still a significant improvement over typical reproducibility limits in conventional disk-type DSC measurements.

This correction strategy was applied to different systems (sapphire, PS, and LA81) with distinct masses, volumes, geometries, and thermal properties, as shown in Fig. 1. Despite these variations in material parameters, the correction consistently achieved a similar precision level of ~1 %, demonstrating that this approach is condition-independent and valid for a broad spectrum of materials.

The correction remained effective across the entire temperature range from 0 °C to 140 °C, as confirmed by measurements performed on PS at several temperatures following the successive pan-repositioning strategy. In each case, $Q_{iso}$ was measured during the isothermal segments preceding $Q_s$ acquisition, and the slope $\frac{dQ_s}{dQ_{iso}}$ was determined from the corresponding empty-pan measurements. The corrected relative error consistently remained close to 1 %, demonstrating that this approach reliably compensates for pan-positioning artifacts across the investigated temperature range. These results highlight the potential of signal post-processing based on $Q_{iso}$ monitoring to enhance measurement accuracy and reproducibility in scanning calorimetry, representing a meaningful step forward in the metrological capabilities of this widely used technique.

### 3.3. Model-Based compensation for pan placement variability

The existence of a non-zero isothermal heat flow ($Q_{iso}$), even after proper sensor calibration and correction for temperature dependence, and its dependence on pan positioning, suggests that additional heat-transfer pathways are active within the DSC furnace (Fig. 5). Under

<figure><img src="image_7.png" /><figcaption>Fig. 5. Simplified electrothermal equivalent circuit of a disc-type heat-flux calorimeter (DSC). The network includes the furnace block at temperature $T_0$; the crucible temperatures $T_1$ (sample) and $T_2$ (reference); and the sensor-disc temperatures $T_3$ (sample) and $T_4$ (reference). Thermal resistances comprise $R_1$ (sensor-base paths) and the pan-sensor contact resistances $R_3$ (reference) and $R_3+\Delta R_3$ (sample). Thermal capacities $C_1, C_2, C_3$, and $C_4$ represent, respectively, the sample and its pan, the reference pan, and the sample and reference side of the sensor disc. An external thermal leakage path is modeled by a reservoir at temperature $T_f$ connected through $R_f$ (reference) and $R_f + \Delta R_f$ (sample) to the corresponding pans. (In the full bridge model, $T_3$ and $T_4$ are cross-coupled by $R_{2,1}$).</figcaption></figure>
realistic conditions, the furnace cannot be considered perfectly adiabatic, and heat losses can occur from the pan to the surrounding environment through convective exchange with the purge gas. This leakage, combined with imbalances in the pan-sensor thermal contact resistance (PS-TCR), appears as a measurable heat flow during isothermal holds, driven by residual temperature differences between the sample and reference pans. Interestingly, the near-zero value of Q<sub>iso</sub> observed at a unique temperature T<sub>f</sub> ≈ -60 °C, regardless of pan position, suggests the existence of a thermal equilibrium point at which no net heat exchange occurs between the pans and the furnace environment (Fig. 3). This reference temperature can represent an effective thermal sink, corresponding to a balance temperature between the cold source imposed by the intracooler (≈ -90 °C) and the ambient environment.

The effect of heat leakage is accounted for by extending the conventional DSC electrical-equivalent thermal model (Fig. 5) to include a cold sink and a thermal resistance representing convective heat exchange with the external environment [31,32]. The leakage pathway is modeled as a connection to an effective cold temperature (T<sub>f</sub> ≈ -60 °C) through the resistances R<sub>f</sub> (reference) and R<sub>f</sub> + ΔR<sub>f</sub> (sample). Pan-positioning variability in a disc-type heat-flux DSC is represented as a change in pan-sensor thermal contact resistance, with R<sub>3</sub> assigned to the reference side and R<sub>3</sub> + ΔR<sub>3</sub> to the sample side. The network response is solved using nodal (matrix) analysis (Equation S1). In the steady isothermal regime, where the heat capacities are considered open, the measurable temperature difference is ΔT = T<sub>3</sub> - T<sub>4</sub>, and the corresponding apparent isothermal heat flow is given by Q<sub>iso</sub>= (T<sub>3</sub> - T<sub>4</sub>)/R<sub>eq</sub>, with 1/R<sub>eq</sub> = 1/R<sub>1</sub> + 2/R<sub>2</sub>, evaluated under no-leakage conditions.

The external leakage is likely governed by convection through the surrounding gas and can be estimated as, R<sub>f</sub> ≈ 1/(h<sub>c</sub> A<sub>top</sub>) with h<sub>c</sub> the convective heat transfer coefficient, typically h<sub>c</sub> ≈ 5-10 W/m<sup>2</sup>/K and A<sub>top</sub> the upper pan area (A<sub>top</sub> ≈ 0.28 cm<sup>2</sup>), yielding R<sub>f</sub> ~ 3600 K/W-7200 K/W. This value is orders of magnitude larger than the other thermal resistances in the DSC system (typically only a few tens of K/W). This pronounced separation of scales justifies the first-order asymptotic approximation used to estimate the apparent heat flow under isothermal conditions, leading to:

$$ \dot{Q}_{iso} \approx \frac{(\Delta R_f + \Delta R_3)}{R_f^2} \cdot (T_0 - T_f) + \dot{Q}_{iso,T_f} \quad (5) $$

This expression captures the observed variation of the measured Q<sub>iso</sub> with pan position, which is reflected in the variation of ΔR<sub>3</sub>. A complete description of the temperature dependence of Q<sub>iso</sub>, as shown in Fig. 3, would require explicit knowledge of the temperature dependence of these resistances, particularly R<sub>f</sub>, ΔR<sub>f</sub> and ΔR<sub>3</sub>, which are not accessible in this study. Nevertheless, between T<sub>f</sub> and 0 °C, Q<sub>iso</sub> displays an approximately linear dependence on (T<sub>0</sub> - T<sub>f</sub>), consistent with the predicted trend and suggesting that the prefactor (ΔR<sub>f</sub> + ΔR<sub>3</sub>) / R<sub>f</sub><sup>2</sup> is only weakly temperature-dependent in this range.

The variability of Q<sub>iso</sub> below 0 °C, shown in Fig. 3, can be used to estimate the variability of the pan-sensor thermal contact resistance (PS-TCR), expressed as ΔR<sub>3</sub>, by attributing all of the observed Q<sub>iso</sub> fluctuations to ΔR<sub>3</sub> in Eq. (5). Measurements performed with flat-bottom pans exhibit a broader spread of Q<sub>iso</sub> values, consistent with their higher range of positioning variability (Fig. 3). In practice, however, even in this less constrained configuration, experienced users are able to keep Q<sub>iso</sub> variations within approximately 120 µW, a difference that remains barely perceptible by visual inspection. In ergofix pans, the Q<sub>iso</sub> variation is typically about 60 µW across all pan positions, providing a more reproducible reference. Taking this value as representative of typical positioning variability, and using R<sub>f</sub> = 3900 K/W in the calculation, ΔR<sub>3</sub> is estimated to be approximately 15 K/W. This variability in PS-TCR can reasonably be attributed to surface heterogeneities at the pan-sensor interface. Surface asperities affect the thermal contact

conductance, which, in this case, is likely dominated by heat conduction through the interstitial air. Assuming the thermal resistance of the interface arises mainly from conduction through air rather than convection, ΔR<sub>3</sub> can be approximated as Δe/(k·A<sub>bottom</sub>), where Δe is the variation in air-gap thickness controlled by the height of surface asperities, k is the thermal conductivity of air, and A<sub>bottom</sub> is the contact area of the pan bottom. This leads to an estimated asperity height of Δe ≈ 10 µm, which is a physically reasonable value for a surface with manufacturing-related roughness [31].

The deviations from linearity observed above 0 °C in Fig. 3 suggest that one or more of the thermal resistances vary with temperature. This is consistent with the expected temperature dependence of the convective heat transfer coefficient (h<sub>c</sub>), which typically follows a power-law relationship of the form h<sub>c</sub> ∝ T<sup>0.25</sup> and would introduce an additional approximate T<sup>0.5</sup> dependence of Q<sub>iso</sub>. A possible temperature dependence of ΔR<sub>f</sub> may also contribute to this effect, although it has not been quantified in the present study.

The Q<sub>s</sub> dependence on pan positioning was accounted for by numerical simulation using a model implemented in a Python environment. The electrical-equivalent thermal circuit shown in Fig. 5 was used, and the nodal equations (Equation S1) were solved iteratively under thermal conditions identical to those of the experiment: an initial isothermal step, a heating ramp at 10 K/min with a modulation amplitude of ΔT = 2 K, and a final isothermal hold. The heat flow was computed iteratively over the entire thermal cycle, yielding Q<sub>iso</sub> once thermal equilibrium was reached. Model parameters, including the thermal resistances (R<sub>1</sub> = 100 K/W, R<sub>2</sub> = 50 K/W) and thermal capacitances (C<sub>3</sub> = 0.1 J/K, C<sub>4</sub> = 0.1 J/K), were selected to be compatible with the temporal response of the DSC used in this study. The heat capacities of the empty pan (C<sub>1</sub> = C<sub>2</sub> = C<sub>p0</sub> = 0.045 J/K) and of the sample (C<sub>s</sub>) were included in the model, with C<sub>s</sub> systematically varied between 0 and 0.025 J/K. R<sub>3</sub> was input between 10 and 20 K/W [31,32].

Central to the analysis is the observation that the simulated enthalpy signal Q<sub>s</sub> exhibits a proportional relationship with the measured isothermal heat flow Q<sub>iso</sub>, closely matching the experimental trend shown in Fig. 4. This strong agreement confirms that the proportionality holds across a broad range of conditions, including variations in R<sub>1</sub>, R<sub>2</sub>, R<sub>3</sub>, ΔR<sub>3</sub>, C<sub>p0</sub>, C<sub>s</sub>, and R<sub>f</sub>. Fig. 4 shows the matching linear fits of Q<sub>s</sub> versus Q<sub>iso</sub> for both empty pans and pans containing sample, using the thermal resistance and capacitance values specified above and adjusting R<sub>f</sub> as the main fitting parameter. Good agreement was obtained for R<sub>f</sub> ≈ 3900-4200 K/W, consistent with values independently estimated from Q<sub>iso</sub> analysis and from convective heat-leakage calculations.

The proportionality dQ<sub>s</sub>/dQ<sub>iso</sub> therefore provides a reliable measure of pan-positioning sensitivity (Fig. 6). As shown in Fig. 6a, dQ<sub>s</sub>/dQ<sub>iso</sub> from experiment agrees well with simulation results over a range of sample heat capacities and for R<sub>f</sub> values between 3500 and 4200 K/W, further supporting the validity of the model. This demonstrates that dQ<sub>s</sub>/dQ<sub>iso</sub> can be used as a robust correction parameter for pan-positioning effects, provided that Q<sub>iso</sub> remains approximately linear with temperature.

However, when Q<sub>iso</sub> deviates from linear temperature dependence, as observed at higher temperatures in Fig. 3, the model's assumption of linear Q<sub>iso</sub> temperature behavior no longer strictly holds. This leads the simulated dQ<sub>s</sub>/dQ<sub>iso</sub> to decrease with increasing temperature, whereas the experimental values remain nearly constant (Fig. 7). In such cases, a more reliable correction is obtained by using the slope of Q<sub>iso</sub> versus temperature (b<sub>iso</sub>) measured at temperatures where Q<sub>iso</sub> remains linear (typically below 0 °C). Using dQ<sub>s</sub>/db<sub>iso</sub> yields excellent agreement between experiment and simulation and provides a better description of the dependence of Q<sub>s</sub> on pan positioning when Q<sub>iso</sub> no longer follows a linear trend with temperature (Fig. 3). Alternatively, Q<sub>iso</sub> measured at a selected reference temperature below 0 °C can still be used as a practical indicator of pan-positioning sensitivity Q̇.

An alternative approach to correct Q<sub>s</sub> is to derive an analytical
<figure><img src="image_8.png" /><figcaption>Fig. 6. (a) dQs/dQiso obtained from the linear fit of Fig. 4 plotted as a function of the total heat capacity of the sample (C<sub>s</sub> = m<sub>s</sub>C<sub>s</sub>) for various materials: sapphire, LA81, and PS (■). Comparison of these experimental results with simulated dQs/dQiso values calculated using the electrical-equivalent thermal model of Fig. 5 and Equation S1. Simulations were performed with R<sub>f</sub> = 100 K/ W, R<sub>2</sub> = 50 K/W, R<sub>3</sub> = 15 K/W, C<sub>3</sub> = C<sub>4</sub> = 0.1 J/K, and pan heat capacities C<sub>1</sub> = C<sub>2</sub> = C<sub>p0</sub> = 0.045 J/K. The variable parameter is dR<sub>3</sub>, and the resulting dQs/ dQiso values are plotted against the leakage resistance R<sub>f</sub> over a range of values that best match the experimental data. (b) shows simulated dQs/dQiso for empty pans (■, C<sub>p0</sub> = 0.045 J/K for sample and reference) and for pans containing a sample with C<sub>s</sub> = 0.01 J/K (●). (■ orange) represents the empirical relation- ship described by Eq. (6), showing good agreement with the simulated trends for the empty pan case and (■ green) for C<sub>s</sub> = 0.01 J/K.</figcaption></figure>

expression describing the dependence of dQs/dQiso and dQs/dbiso on the thermal parameters of the DSC cell. This was achieved by systematically varying one parameter at a time in the simulation while keeping the others constant, first establishing the linear dependence of dQs/dQiso on Rf and then approximating the contribution of R1, R2, R3, C0, and Cs. This procedure led to an empirical relation, given in Eq. (6), that captures the combined effects of pan and sample heat capacities (Cp0 and Cs) and the leakage resistance Rf, together with a second-order correction account- ing for R1, R2, and R3.

Comparison with the full numerical simulation shows excellent agreement, confirming the validity of this empirical formulation as a practical and simplified description of the pan-positioning correction. Notably, even when the contributions of R<sub>1</sub>, R<sub>2</sub>, and R<sub>3</sub> are omitted, retaining only the terms involving C<sub>p0</sub>, C<sub>s</sub>, and R<sub>f</sub>, the deviation in the estimation of dQs/dQiso and dQs/dbiso remains within ~ 4 %, which is negligible for the purpose of this validation. This demonstrates that the main contribution to the correction can be captured using only quanti- fiable parameters such as pan and sample heat capacities and R<sub>f</sub>.

<figure><img src="image_9.png" /><figcaption>Fig. 7. Effect of temperature on the measured correction parameters dQs/dQiso and dQs/dbiso for empty pans with a mass difference below 0.04 mg. (■) Experimental dQs/dQiso values (left axis). (□) Simulated dQs/dQiso values ob- tained using the parameters described in the text, with R<sub>f</sub> fixed at 3900 K/W and thermal resistances and capacities as specified (left axis). Q<sub>iso</sub> values were taken from the isothermal steps preceding the temperature scans. (●) Experi- mental dQs/dbiso values (right axis), where b<sub>iso</sub> is the slope of Q<sub>iso</sub> versus temperature taken during the isothermal steps prior to 0 °C. (O) Simulated dQs/ dbiso values (right axis), calculated using the same procedure as for dQs/dQiso but using b<sub>iso</sub> as the input parameter.</figcaption></figure>

$$
\frac{dQ_s}{d\dot{Q}_{iso}} \approx 1.93 \frac{\Delta T}{(T_0 - T_f)} (C_{p0} + C_s) \cdot R_f + \frac{1}{(T_0 - T_f)} f(C_{p0}, C_s, R_1, R_2, R_3) \quad (6)
$$

$$
\frac{dQ_s}{db_{iso}} \approx 1.93 \Delta T \cdot (C_{p0} + C_s) \cdot R_f + f(C_{p0}, C_s, R_1, R_2, R_3) \quad (7)
$$

With: f(C<sub>p0</sub>, C<sub>s</sub>, R<sub>1</sub>, R<sub>2</sub>, R<sub>3</sub>) ≈ 262.2 – 323.4 C<sub>p0</sub> – 202.8 C<sub>s</sub> – 0.33 R<sub>1</sub> + 0.078 R<sub>2</sub> – 0.15 R<sub>3</sub>

These results suggest that the contribution of pan positioning to the
enthalpy signal can be described through the combined effect of the PS-
TCR (ΔR<sub>3</sub>) and the leakage resistance (R<sub>f</sub>) as:

$$
Q_s \approx C_s \cdot \Delta T + \Delta C_{p0} \cdot \Delta T + \left[ \begin{aligned}
& 1.93 \cdot (C_{p0} + C_s) \cdot \Delta T \cdot \frac{1}{R_f} \\
& + f(C_{p0}, C_s, R_1, R_2, R_3) \cdot \frac{1}{R_f^2} \end{aligned} \right] \Delta R_3 \quad (8)
$$

In this formulation, the value of R<sub>f</sub> plays a central role and must be
determined for each instrument, as it is likely machine-dependent. R<sub>f</sub>
can be estimated from empty-pan measurements using either dQ<sub>s</sub>/db<sub>iso</sub>
or dQ<sub>s</sub>/dQ<sub>iso</sub>. For instance, using :

$$
R_f = \frac{\left( \frac{dQ_s}{db_{iso}} - p \right)}{(1.93 \Delta T C_{p0})} \quad \text{with } p = f(C_{p0}, 0, R_1, R_2, R_3) \approx 220 \text{ s K} \qquad (9)
$$

Taking dQs/dbiso=900 s K for the empty pan yields R<sub>f</sub> ≈ 3900 K/W,
in good agreement with the value recovered from numerical simulation.
Alternatively, R<sub>f</sub> can be determined from dQs/dQiso at 0 °C for the empty
pan taking the measured dQs/dQiso= 15 s and

$$
R_f = \frac{\left( \frac{dQ_s}{d\dot{Q}_{iso}} - \frac{p}{(T_0 - T_f)} \right)}{(1.93 \Delta T C_{p0})} (T_0 - T_f) \quad (10)
$$

which gives R<sub>f</sub> ≈ 3937 K/W. Even when neglecting the contributions of
R<sub>1</sub>, R<sub>2</sub>, and R<sub>3</sub>, which may not be known to the user, the resulting R<sub>f</sub>
remains ≈ 3756 K/W, introducing <4 % deviation in dQ<sub>s</sub>/dQ<sub>iso</sub> and dQ<sub>s</sub>/
db<sub>iso</sub>, which is negligible for practical correction purposes.

The above results show that sample-positioning effects can also be
corrected using the Eq. (4), provided that the dependence of dQs/dQiso
and dQs/dbiso on pan positioning is known for the empty pan. This
requires measuring Qs versus Qiso for the empty pan and extrapolating the curve to obtain Qiso,0, the isothermal value corresponding to zero Qs. Using this reference, a corrected enthalpy signal Qs can be calculated and subsequently used to determine the absolute cP. For improved accuracy, the influence of the sample heat capacity Cs on dQs/dQiso and dQs/dbiso can be incorporated empirically. These dependencies can be approximated as

$$ \left(\frac{dQ_s}{dQ_{iso}}\right)_{sample} \approx \alpha \left(1 + \frac{C_s}{c_{P0}}\right) - \beta \frac{C_s}{c_{P0}} \quad \text{with } \alpha = \left(\frac{dQ_s}{dQ_{iso}}\right)_{empty pan} \quad \text{and} \\ \beta \approx \frac{f(C_{P0}, C_s, R_1, R_2, R_3)}{(T_0 - T_f)} \approx 3.6 \text{ s} $$

Or

$$ \left(\frac{dQ_s}{db_{iso}}\right)_{sample} \approx \gamma \left(1 + \frac{C_s}{c_{P0}}\right) - \delta \frac{C_s}{c_{P0}} \quad \text{with } \gamma = \left(\frac{dQ_s}{db_{iso}}\right)_{empty pan} \\ \text{and } \delta \approx f(C_{P0}, C_s, R_1, R_2, R_3) \approx 220 \text{ s K} \qquad (11) $$

### 3.4. Strategies for pan-placement compensation in cP determination

The uncertainty budget for the heat capacity (cP) measurement identifies the variability of the integrated heat flow (Qs) across the requisite three measurements as the dominant contribution to the overall error, particularly for weak-signal samples. To improve measurement precision, two strategies targeting this key variable were investigated. The first strategy focuses on a priori control through meticulous pan placement. By maintaining a fixed reference pan position and carefully repositioning the sample pan to achieve a consistent isothermal heat flow (Qiso), the contribution from positional variation to the uncertainty budget can be minimized. In this work, the magnitude of this contribution was characterized by deliberately shifting the sample pan within the sensor's central zone and measuring the resultant Qiso at 0 °C after thermal equilibrium. Through careful manual placement, the variation (δQiso) was reduced to approximately 10 µW optimally and 20 µW typically, which limited the variability in Qs to ≈ 0.3 mJ. This corresponds to a propagated uncertainty component in cP of about 4.2 %.

A more robust second strategy employs a posteriori analytical correction. The method involves constructing an a priori curve of enthalpy signal Qs versus Qiso to determine the sensitivity slope, $\frac{dQ_s}{dQ_{iso}}$. An initial correction can be performed using the slope obtained from an empty pan, as defined in Eq. (1) (Fig. 1, ▲ green). An example of the uncertainty budget derived with this approach is presented in Table 1. The table contrasts the conventional method, treating pan repositioning as a random error and propagating uncertainties via Eqs. (1)–(3), with a Qiso referenced baseline-subtraction scheme that explicitly accounts for position-dependent errors through Qiso (uncertainties propagated on ΔQs evaluated at matched Qiso). However, for superior accuracy, an enhanced method (Eq. (11)) utilizes the sample-specific slope $\left(\frac{dQ_s}{dQ_{iso}}\right)_{sample}$, which inherently accounts for the heat capacity of the sample itself (Fig. 1, –). Alternatively, $\left(\frac{dQ_s}{dQ_{iso}}\right)_{sample}$, can be deduced from multiple measurements performed at different sample positions (Fig. 1, *). At this stage, further error reduction is not strictly necessary, as other contributors to the overall uncertainty budget, such as the thermal conductivity of the sample, its spatial distribution within the pan, and mass measurement precision, become dominant. Nevertheless, applying either of these last two correction approaches, combined with a precisely known sample mass and a fixed reference pan, reduces the corresponding error contribution and yields a final cP uncertainty slightly below 1 %.

Application of this step-scan strategy for specific heat capacity (cP) determination was evaluated on aluminum specimens (mass 7.95 mg; integrated enthalpy Qs ≈ 13.7 mJ), as shown in Fig. 8. The empty pan (used without a lid) was measured ten times to construct a calibration of Qs versus Qiso, enabling accurate zero-line subtraction of the sample-pan

<caption>**Table 1**</caption>

Uncertainty budget for the specific heat capacity cP of Laropal A81 determined by DSC. The first budget follows the standard baseline-subtraction approach using Eqs. (1)–(3). The budget is shown for the Qiso referenced baseline subtraction, where ΔQs = Qs(Qiso) – Qs,0(Qiso) and ΔQs,saph = Qs,saph(Qiso) – Qs,0(Qiso). In this second case, uncertainties are propagated directly on the subtracted signals ΔQs rather than via Eq. (2). Here, Qs(Qiso) denotes the heat-flow values evaluated at selected Qiso positions.

<table><caption>**Uncertainty Budget for c<sub>p</sub> using the method described in the text**</caption><caption>Uncertainty Budget for c<sub>p</sub> using standard method</caption><thead><tr><th>Quantity</th><th>Symbol</th><th>Value</th><th>Unit</th><th>Std. uncertainty (absolute)</th></tr></thead><tbody><tr><td>Sample heat flow</td><td>Q<sub>s</sub></td><td>20.82</td><td>mJ</td><td>0.86</td></tr><tr><td>Sample Baseline</td><td>Q<sub>s,0</sub></td><td>1.17</td><td>mJ</td><td>0.72</td></tr><tr><td>Sapphire heat flow</td><td>Q<sub>s,saph</sub></td><td>34.87</td><td>mJ</td><td>0.60</td></tr><tr><td>Sapphire baseline</td><td></td><td>-1.04</td><td>mJ</td><td>0.60</td></tr><tr><td>Sample mass</td><td>m<sub>sample</sub></td><td>7.54</td><td>mg</td><td>0,01</td></tr><tr><td>Sapphire mass</td><td>m<sub>saph</sub></td><td>25.61</td><td>mg</td><td>0,01</td></tr><tr><td>ΔQ<sub>s</sub></td><td>Q<sub>s</sub> - Q<sub>s,0</sub></td><td>19.65</td><td>mW</td><td>1,12</td></tr><tr><td>ΔQ<sub>s,saph</sub></td><td>Q<sub>s,saph</sub> - Q<sub>s,0</sub></td><td>35.91</td><td>mW</td><td>0.85</td></tr><tr><td>Heat capacity</td><td>c<sub>p</sub></td><td>1.33</td><td>J g<sup>-1</sup> K<sup>-1</sup></td><td></td></tr><tr><td>Relative uncertainty</td><td>(δc<sub>p</sub>/c<sub>p</sub>)</td><td>0,06</td><td>1σ, relative</td><td></td></tr></tbody></table>

<table><tbody><tr><td>ΔQ<sub>s</sub></td><td>Q<sub>s</sub>(Q<sub>iso</sub>) - Q<sub>s,0</sub><br>(Q<sub>iso</sub>)</td><td>19.62</td><td>mW</td><td>0.15</td></tr><tr><td>ΔQ<sub>s,saph</sub></td><td>Q<sub>s,saph</sub>(Q<sub>iso</sub>) - Q<sub>s,0</sub><br>(Q<sub>iso</sub>)</td><td>35.98</td><td>mW</td><td>0.1</td></tr><tr><td>Heat capacity</td><td>c<sub>p</sub></td><td>1.330</td><td>J g<sup>-1</sup> K<sup>-1</sup></td><td></td></tr><tr><td>Relative uncertainty</td><td>(δc<sub>p</sub>/c<sub>p</sub>)</td><td>0.008</td><td>1σ, relative</td><td></td></tr></tbody></table>

<figure><img src="image_10.png" /></figure>

**Fig. 8.** Specific heat capacity, cP, of the aluminum standard (7.95 mg) measured by step-scan between 20 and 22 °C using 10 K/min ramps, ΔT = 2 K per step, and 3 min isotherms. The sequence involved repeated repositioning of the empty pan, the sample pan, and the sapphire standard (25.64 mg). (▲ red) cP values obtained by enthalpy-based zeroline subtraction per Eq. (4), using the empty-pan dependence of Qiso to set the reference position. (—) documented cP at 20 °C [22]. (—) mean cP from the present measurements. (—, red) cP using second, independent run of the sapphire standard processed with the same Qiso based subtraction. For comparison, cP values from a single-test step-scan sequence (empty pan → sample → empty pan → sapphire reference) are also shown (◊).

signal via Eq. (4). Step-scan experiments were conducted between 20 °C and 22 °C following the procedure described above. Sample signals acquired over multiple repeats were corrected using the Qiso based subtraction, normalized by a sapphire reference processed identically, and reported as ▲ red. The resulting cP values exhibit a dispersion below 1 % and agree with the documented aluminum cP within 1 % [22]. By contrast, a conventional step-scan workflow relying on single runs of empty pan → sample → empty pan → sapphire, with pan repositioning between runs, produced cP dispersions exceeding 10 %, which is expected given the small aluminum mass and the correspondingly low enthalpy levels (≈ 13–14 mJ). These results demonstrate that Qiso anchored empty-pan subtraction (Eq. (4)) substantially improves cP
precision and repeatability, even for sample masses well below instru-
ment recommendations.

4. Conclusion

This study addresses the decades-old challenge of systematic error in
absolute heat capacity measurements within disk-type heat-flux DSC,
originating from manual pan repositioning and the associated variability
of pan-sensor thermal contact resistance (PS-TCR). It shows that PS-TCR
variability, a perennial source of error, can be repurposed from an
irreproducible artifact into a measurable and correctable signature. We
demonstrate that the isothermal heat flow (Qiso), recorded during step-
scan holds, serves as a direct quantitative proxy for this variability. By
establishing a robust linear correlation between Qiso and the integrated
enthalpy signal (Qs), we develop a practical correction procedure that
effectively isolates and subtracts the positioning-dependent bias.

This methodology is supported by an extended electro-thermal
model incorporating a weak leakage path to an effective surrounding
sink temperature (Tf), which explains the non-zero isothermal heat flow
and yields realistic leakage resistance values (Rf). When applied to a
range of materials, including polymers, sapphire, and low-signal sam-
ples, the protocol systematically reduces cₚ measurement uncertainty to
below 1 % for 5 mg samples, without hardware modification. Crucially,
the method promotes Qᵢₐₛ as an in-situ quality metric for thermal contact
symmetry and delivers a software-only correction, transferable between
instruments via empty-pan calibration. By transforming a fundamental
limitation into a calibration tool, this approach makes high-accuracy
absolute cₚ determination on conventional disk-type DSC instruments
broadly accessible.

CRediT authorship contribution statement

Yahya Rharbi: Writing – review & editing, Writing – original draft, Visualization, Validation, Supervision, Software, Resources, Project administration, Methodology, Investigation, Funding acquisition, Formal analysis, Data curation, Conceptualization. Frédéric Hugenell: Visualization, Validation, Resources, Methodology, Investigation, Formal analysis, Data curation, Conceptualization.

Declaration of competing interest

The authors declare that they have no known competing financial
interests or personal relationships that could have appeared to influence
the work reported in this paper.

Acknowledgments

LGP2 is part of the LabEx Tec 21 (Investissements d'Avenir – grant agreement n°ANR-11-LABX-0030) and of PolyNat Carnot Institutes Investissements d'Avenir – grant agreement n°ANR-16-CARN-0025-01). The Laboratoire Rhéologie et Procédés is part of the Labex Tec 21 (Investissement d'Avenir – grant agreement n° ANR-11-LABX-0033) and of the Polynat Carnot Institut (Investissement d'Avenir – grant agreement n°ANR-11-CARN-030-01). The authors gratefully acknowledge Eric Faivre for his assistance in conducting the experiments.

Supplementary materials

Supplementary material associated with this article can be found, in
the online version, at doi:10.1016/j.tca.2025.180192.

Data availability

Data will be made available on request.

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