Thermal transport in non-metal materials is mediated by phonons and follows Fourier diffusion theory.7 However, when the physical size of the material approaches its phonon mean free path, the thermal gradient between regions at different temperatures disappears (Fig. 1) and phonons enter a ballistic movement regime, described by the non-diffusive Boltzmann transport equation.6,7,5 Non-diffusive heat transport can lead to a substantial reduction of the phonon thermal conductivity of the material, resulting in erroneous absolute temperature and heat transport predictions when employing bulk thermal parameters.8 Although non-diffusive thermal transport has been observed in nanoparticles,9,10 nanoparticle disordered films,11 and nanowire arrays,12 studying thermal dynamics and discriminating between heat propagation regimes remains a big challenge, especially for nanofluids. Current techniques including time-domain thermoreflectance,13 scanning thermal microscopy,14 transient thermal grating,15 and other microscale imaging techniques16 are relatively complex and limited only to surface probing. Meanwhile, the accuracy of theoretical approaches based on molecular dynamics simulations is contingent on the choice of empirical interatomic potentials, and ab initio simulations often examine phonons at 0 K rather than at finite temperatures.17

Luminescence thermometry is a powerful yet simple technique used to investigate the thermal behavior of nanofluids,18,19 heat transfer across cellular barriers,20,21 and the intracellular medium.3,22−24 As a class of luminescent nanothermometers, UCNPs can report temperatures with high spatiotemporal and thermal accuracy, easily form colloidal solutions and are amenable to solution-processing.25–27 Notably, the core/shell engineering of UCNPs affords rational placement of light upconverting Ln3+ dopants in distinct layers for purposefully-built heterostructures. To investigate heat propagation within the UCNPs themselves, here we exploit core/shell engineering and the temperature-sensitivity of different Ln3+ ions by creating luminescent nanothermometers with spatially-delayed thermal response to an external heating. Our UCNPs provide in situ transient thermal recordings that allow to estimate the heat propagation speed in the NaGdF4 matrix and to discriminate between diffusive and non-diffusive heat transport regimes occurring in the nanoparticles.

Luminescent thermometers with spatially-delayed temperature sensing were synthesized by hot-injection thermal decomposition which facilitates layer-by-layer growth of the UCNPs and allows to place Ln3+ dopants at the designated positions, henceforth thermometric layers (Supplementary Section I, Figs. 1–14). Er3+,Yb3+ and Tm3+,Yb3+ co-dopants were rationally introduced in the NaGdF4 host (Fig. 1a), comprising three distinct architectures with: (i) a single thermometric layer ~ 28 nm away from the surface (**Er1** and **Tm1**), (ii) ~ 5 nm from the surface (**Er2** and **Tm2**), and (iii) two thermometric layers separated by ~ 28 nm (**Er1-Tm2**). Thin Ln3+-doped thermometric layers ensure more uniform heat wave detection, and non-overlapping upconversion emissions of Er3+ and Tm3+ allow for simultaneous temperature monitoring under near-infrared laser irradiation in our custom-built setup (Fig. 2a). To further simplify the analysis of the thermal recordings we studied ligand-free UCNPs as water-suspended nanofluids.

Under continuous 980 nm laser excitation, the Yb3+, Er3+-doped UCNPs display the characteristic upconversion emission of Er3+ with green (2H11/2/4S3/2→4I15/2) and red (4F9/2→4I15/2) radiative transitions at around 525/545 and 660 nm, respectively (Fig. 2b). The emission intensity ratio from the thermally-coupled green (2H11/2 and 4S3/2) excited states is used to calibrate the **Er1**, **Er2**, and **Er1-Tm2** luminescent thermometers (Supplementary Figs. 15–17). Their relative thermal sensitivity (*S*r) and temperature uncertainty (*δT*) are .3 %·K− 1 and 1.0 K, respectively (Supplementary Fig. 18).27 UCNPs containing Tm3+, Yb3+-doped thermal layers upconvert 980 nm excitation to blue (1D2→3F4, 1G4→3H6) and red (1G4→3F4) emissions. The intensity ratio of the blue emission bands from both 1D2 and 1G4 excited states is used to calibrate the **Tm1**, **Tm2**, and **Er1-Tm2** luminescent thermometers (Supplementary Fig. 17), having *S*r~0.14 %·K− 1 and *δT* ~ 0.8 K for an acquisition time of 1250 ms (Supplementary Fig. 18). Details on the calibration and thermal uncertainty are presented in Supplementary Section II.

To study heat propagation dynamics in nanoparticles, a UCNP nanofluid was subjected to a temperature gradient induced by a heating contact. The experimental setup allows to set the magnitude of the temperature step and to record transient thermal curves with an adjustable integration time. Transient heating curves of **Er1**/**Er2** (Figs. 3a, b) and **Tm1**/**Tm2** (Figs. 3c, d) have a typical functional form of an external heating process in nanofluids.18,28−30 We can observe that the temperature increase is not recorded immediately with the instantiation of heating, and the onset time *t*0 at which the temperature change is registered depends on the spatial placement of the thermometric layer within the UCNP (Supplementary Section III). As expected, *t**0* values are greater for core-adjacent thermometric layers of UCNP (**Er1** and **Tm1**) than for the surface-proximal ones (**Er2** and **Tm2**). However, the onset times between the two architectures differ by as much as ≈ 20 s. We note that the reported effects are fully reversible and independent of the setup, experimental variables, or temperature-sensitive Ln3+ ions (Er3+ or Tm3+) (Supplementary Figs. 19–21).

We can extract the effective heat transfer velocity (\({v}_{h}\)) as the slope of the *t**0* vs. the location of the thermometric layers within the UCNPs (Fig. 4a). Regardless of the Ln3+ ion employed for luminescence thermometry we estimate \({v}_{h}=1.3\pm 0.1\) nm s−1, four orders of magnitude slower than expected by Fourier law (Supplementary Section III). Note that *t*0 does not depend on the employed Joule heating rate (Supplementary Fig. 21).

Next, we recorded the transient heating curves from **Er1-Tm2** UCNPs. The two thermometric layers placed away from each other in **Er1-Tm2** UCNPs registered a temperature increase in the nanofluid at different onset times. The *t**0* of the Tm3+ thermometric layer in **Er1-Tm2** is similar to **Er2** and **Tm2**, and lower than *t**0* of the Er3+ one. The latter was also greater than **Er1** and **Tm1**, due to the larger total size of the **Er1-Tm2** UCNPs (Fig. 1a). The linear dependence of *t*0 with the distance travelled across the **Er1-Tm2** UCNP corroborates constant \({v}_{h}\) in NaGdF4 UCNPs, no matter the Ln3+ doping (Fig. 4a).

Interestingly, while the core-adjacent Er3+ thermometric layer in **Er1-Tm2** detects the transient heating profile similar to **Er1** and **Tm1**, the Tm3+ thermometric layer manifests in a more complex behavior (Fig. 4b**)**. After registering heating of the nanofluid (*t**0* = 10 s), Tm3+ ions in the surface-proximal thermometric layer of **Er1-Tm2** report a temperature increase for the first 5 s, followed by a 20 s plateau. Subsequently, the temperature increases again, until a steady-state is reached. We have calculated the diffusive heat propagation length, \({v}_{h}t\), and found a perfect match between the observed temperature changes and the dimensions of the distinct layers of **Er1-Tm2** (Figs. 4c,d). Each discontinuity in the temperature increase, registered by the Tm3+ thermometric layer, corresponds to a unique compositional section of the **Er1-Tm2** as the heat wave crosses UCNPs. This unequivocally validates that the effective heat wave velocity is constant and invariant to UCNPs composition or the interfaces of the core/shell structures.

We further analyze the transient heating profile measured in **Er1-Tm2** as a thermal gradient \({\nabla }_{r}T=\frac{{T}_{1}-{T}_{2}}{L}\) (Supplementary Section IV); here, *T*1 and *T*2 are the temperatures respectively recorded by the Tm3+ and Er3+ thermometric layers, and *L* is the mean distance between them. Since the heat transfer regime is determined by the relationship between the size of a nanomaterial and its phonon mean free path, it is convenient to normalize the thermal gradient against *r/L*, where \(r={v}_{h}{t}_{h}={v}_{h}\left(t-{t}_{0}\right)\) is the distance travelled by the heat way (*t* being the heating elapse time). Thermal gradients in the diffusive regime are then considered within the framework of the lumped elements’ thermal circuit,31 in which the heat flow is a thermal current (*i*), while the heat transfer by diffusion maps onto the thermal resistance (*R*) and capacitances (*C*) of each layer. In the simplified thermal circuit of **Er1-Tm2**, *R*1/*C*1 and *R*2/*C*2 are used to describe the thermal gradients in the Tm3+ and Er3+ thermometric layers, respectively (Fig. 5a).

Four distinct regions were found in \({\nabla }_{r}T\) (Fig. 5b). In the \(r/L<1\) region, there is no thermal gradient within the UCNPs (Supplementary Fig. 22a), despite temperature increase registered in both thermometric layers, indicative of the non-diffusive heat transport between them. The non-diffusive regime is active until the heat wave reaches the core-adjacent Er3+ thermometric layer, since \(r<L\). In the second region, \(1<r/L<3\), a positive thermal gradient within the UCNP (Supplementary Fig. 22b) represents thermal loading of *C*1. In \(1<r/L<2\) the thermal current flow in *C*1 (denoted by *i*1C) is greater than in *R*1 (denoted by *i*1R), thus the energy is transported by *i*1 (heating from the surrounding water) and loaded into *C*1. For \(2<r/L<3\), the \({\nabla }_{r}T\) approaches zero as *C*1 is maximally loaded and the thermal current begins to flow in Er3+ thermometric layer. The null thermal gradient is reached at \(r/L=3\). The third region, \(3<r/L<7.5\), displays a negative thermal gradient (Supplementary Fig. 22c). Within \(3<r/L<4.5\) thermal current *i*1 is used to load *C**2* and when passing into \(4.5<r/L<7\) the thermal gradient tends to zero as *C*2 reaches its maximum load. Finally, we observe a null thermal gradient for \(r/L>7.5\) (Supplementary Fig. 22d) when both thermometric layers are recording the same steady-state temperature, meaning that the system has reached a net-zero balance between the ingoing and outgoing heat fluxes. From the point-of-view of thermal capacitance, maximum thermal loading is attained in both *C**1* and *C**2*, and the thermal current flows only through *R*1 and *R*2.

The thermal loading of Tm3+ and Er3+ thermometric layers of **Er1-Tm2** can be further supported by an analogous analysis of a hypothetical nanoparticles with two Tm3+ layers as in **Tm1-Tm2** or two Er3+ layers in **Er2-Er1** (Supplementary Section IV). Experimental transient heating profiles independently recorded by **Tm1**, **Tm2**, **Er1** and **Er2** (Figs. 3a-d) were used to calculate the temperature gradients in the hypothetical **Tm1-Tm2** and **Er2-Er1** (Supplementary Fig. 23). For both simulated geometries, we observe a null thermal gradient in \(r/L<1\), clearly attesting to the non-diffusive heat transport in the nanoparticles and it being invariant to the composition of the thermometric layers. Individually, the thermal loading of the surface-proximal Tm3+ thermometric layer in **Tm1-Tm2** (Supplementary Figs. 23a,b) occurs for \(1<r/L<2\), and that of the core-adjacent Er3+ thermometric layer in **Er2-Er1** for \(1<r/L<2.5\) (Supplementary Figs. 23c,d). Although maximum values of \({\nabla }_{r}T\) between Tm3+ thermometric layers in **Tm1-Tm2** and **Er1-Tm2** differ due to the greater thermal capacity of the the latter structure (Supplementary Table 6), the \({\nabla }_{r}T\) vs \(r/L\) dynamics between the two nanoparticles are in near-perfect agreement. Comparing **Er2-Er1** and **Er1-Tm2**, the minimum \({\nabla }_{r}T\) and thermal loading dynamics are very similar as the physical dimensions of the core-adjacent Er3+ thermometric layers are the same.

The thermal loading of each thermometric layer of **Er1-Tm2** can be further described by the characteristic time \({\tau }_{j}={R}_{j}{C}_{j}\),32 where *j* = 1,2 correspond to Tm3+ and Er3+, respectively. The thermal loading times were determined by fitting the experimental data (solid lines in Fig. 5b) to a diffusive heat propagation model (Supplementary Eq. 9). We estimate τ1 = 20 ± 4 s and τ2 = 18 ± 3 s (Supplementary Table 5), which are comparable, within experimental error, to the characteristic times extracted from the simulated geometries, τ’1=21±1 s (for **Tm1-Tm2**) and τ’2=19±1 s (for **Er2-Er1**). The characteristic times are the same with (in **Er1-Tm2**) and without (in the hypothethical architectures) thermal exchange between the embedded thermometric layers as both thermal capacitances are being loaded simultaneously under a low heating rate (< 10 W·cm−2). To affirm that the diffusive heat transport explains thermal loading of the thermometric layers when \(r/L>1\), we determine the expected values of \({\tau }_{j}\) considering the dimensions and compositions of the the respective layers (Supplementary Table 6). The predicted values are τ1 = 21±8 s, τ2 = 21 ± 8 s, τ’1=19±8 s, and τ’2=22±9 s and within the uncertainty match the experimentally obtained ones. Thus, the existence of different thermometric layers exchanging energy as heat does not affect the diffusive heat exchange within the UCNPs and follows the classical Fourier law. However, the overall thermal gradient profile can be altered in complex heterostructures, as the additional layers accumulate and store the thermal energy.

In brief, we have accurately quantified the heat transport taking place in UCNPs unveiling the presence of non-diffusive and diffusive regimes inside the nanoparticles. We have measured the relatively slow propagation of the heat wave within the nanoparticles at the speed of around 1.3 nm∙s− 1 which is, remarkably, four orders of magnitude slower than the value predicted by Fourier’s law. Moreover, the heat propagation between the distinctly placed thermometric layers is primarily non-diffusive before a thermal gradient is established. Furthermore, depending on the structure of the nanoparticles, the diffusive heat transport assumes different normalized heat propagation lengths and shows differential thermal loading in the same nanoparticle. Our results pave the way to extract phenomenological heat propagation parameters at the nanoscale through a generalized use of luminescence nanothermometry, which will become instrumental in the nanotechnological development of optoelectronics,33 catalysis,34 thermal biology,23 and precision medicines.35