Differential scanning calorimetry

Primary thermogram $\Phi(T) = \frac{dQ}{dt}$ Exothermic Peak

Baseline:

Heat Flow $\Phi(T)$ Baseline $\Phi_{\text{base}}(T)$ Onset tangent $\left.\frac{d\Phi}{dT}\right|_{\max}$ Pre-Transition Tangent

Hover to read $(T, \Phi)$

Start Boundary ($T_1$): 110.0 °C

Pre-transition bound $\Phi(T_1): 0.120\text{ mW}$

End Boundary ($T_2$): 165.0 °C

Post-transition return $\Phi(T_2): 0.080\text{ mW}$

$\alpha(T) = \frac{\int_{T_1}^T \Delta\Phi(T') dT'}{\int_{T_1}^{T_2} \Delta\Phi(T') dT'}$

1. Non-Isothermal Transformation Kinetics & The Peak Shift

During differential scanning calorimetry at a constant programmed heating rate $\beta = \frac{dT}{dt}$, the rate of phase transformation $\frac{d\alpha}{dt}$ obeys Arrhenius-governed solid-state rate laws:

$$\frac{d\alpha}{dt} = k(T) \cdot f(\alpha) = A \exp\left(-\frac{E_a}{R T}\right) \cdot f(\alpha)$$

Because real time $\Delta t = \frac{\Delta T}{\beta}$ shrinks as the heating rate $\beta$ is increased, a higher temperature is required to supply sufficient thermal activation energy. Consequently, the transformation peak apex $T_p$ systematically shifts to higher temperatures according to the Kissinger condition:

$$\ln\left(\frac{\beta}{T_p^2}\right) = -\frac{E_a}{R T_p} + \ln\left(\frac{A R}{E_a}\right)$$

Additionally, furnace-to-crucible thermal resistance $R_{\text{th}}$ creates an instrumental thermal lag:

$$\Delta T_{\text{lag}} = T_{\text{program}} - T_{\text{sample}} \approx R_{\text{th}} \cdot C_{\text{total}} \cdot \beta$$

2. ASTM E794 / ISO 11357 Extrapolated Onset Determination

The extrapolated onset temperature $T_{\text{onset}}$ corresponds to the intersection between the pre-transition baseline and the tangent constructed at the inflection point of steepest leading edge slope:

$$T_{\text{onset}} = T_{\text{infl}} - \frac{\Phi(T_{\text{infl}}) - \Phi_{\text{base}}(T_{\text{infl}})}{\left.\frac{d\Phi}{dT}\right|_{\max} - \left.\frac{d\Phi_{\text{base}}}{dT}\right|_{T_1}}$$

3. Baseline Demarcation Models & Crystallinity

When a heat capacity step $\Delta C_p$ exists across the transformation (e.g. glass transitions or semi-crystalline polymer melting), a sigmoidal baseline continuously updates the baseline position via the fractional conversion $\alpha(T)$:

$$\Phi_{\text{base}}(T) = \Phi_{\text{pre}}(T) + \alpha(T) \cdot \left[\Phi_{\text{post}}(T) - \Phi_{\text{pre}}(T)\right]$$

The observed enthalpy $\Delta H$ and degree of crystallinity $X_c$ are calculated as:

$$\Delta H = \frac{1}{m \cdot \beta} \int_{T_1}^{T_2} |\Phi(T) - \Phi_{\text{base}}(T)| \, dT, \qquad X_c = \frac{\Delta H}{\Delta H^\circ} \times 100\%$$

Conversion: $\alpha = 0 \to 1.00$ Active Heating Rate: $\beta = 10.0\text{ }^\circ\text{C/min}$

Operational parameters

Live Solver

Heating rate ($\beta = \frac{dT}{dt}$): 10.0 °C/min

$2.0\text{ }^\circ\text{C/min}$ $25.0\text{ }^\circ\text{C/min}$ $50.0\text{ }^\circ\text{C/min}$

• Notice that shifting $\beta$ from 2 to $50\text{ }^\circ\text{C/min}$ pushes $T_p$ rightward by both Arrhenius kinetics and $\Delta T_{\text{lag}}$.

Sample mass ($m$): 5.00 mg

$1.0\text{ mg}$ (Thin pan film) $25.0\text{ mg}$ (Bulk specimen)

Thermal resistance ($R_{\text{th}}$): 0.08 K·s/mJ

$0.01\text{ K}\cdot\text{s/mJ}$ (Ideal) $0.25\text{ K}\cdot\text{s/mJ}$ (High lag)

Reference 100% enthalpy ($\Delta H^\circ$):

J/g

Construct standardised inflection tangents Simulate crucible thermal lag ($\Delta T_{\text{lag}} = R_{\text{th}} \cdot C_{\text{tot}} \cdot \beta$)

Thermodynamic metrics

Extrapolated Onset ($T_{\text{onset}}$) Tangent intersection

127.42 °C

Peak apex ($T_{\text{peak}}$) $\left.\frac{d\Phi}{dT}\right|_{T_p} = 0$

134.80 °C

Extrapolated endset ($T_{\text{end}}$) Baseline return intercept

146.10 °C

Thermal energy ($Q$) $Q = \frac{1}{\beta}\int \Delta \Phi \, dT$

240.00 mJ

Specific enthalpy ($\Delta H$) $\Delta H = \frac{Q}{m}$

48.00 J/g

Crystallinity ($X_c$) $X_c = \frac{\Delta H}{\Delta H^\circ} \times 100\%$

34.3 %

Half-height width ($\text{FWHM}$) Peak broadening criterion

7.60 °C