Spatial Energy Transfer and Migration Model for Upconversion

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C: Physical Processes in Nanomaterials and Nanostructures

Spatial Energy Transfer and Migration Model for Upconversion Dynamics in Core-Shell Nanostructures Jiangfan Liu, Tairan Fu, and Congling Shi J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b12300 • Publication Date (Web): 13 Mar 2019 Downloaded from http://pubs.acs.org on March 21, 2019

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The Journal of Physical Chemistry

Spatial Energy Transfer and Migration Model for Upconversion Dynamics in Core-Shell Nanostructures Jiangfan Liu1, Tairan Fu1,* , Congling Shi2,# 1. Key Laboratory for Thermal Science and Power Engineering of Ministry of Education, Beijing Key Laboratory of CO2 Utilization and Reduction Technology, Department of Energy and Power Engineering, Tsinghua University, Beijing 100084, P. R. China 2. China Academy of Safety Science & Technology, Beijing 100029, P.R. China Corresponding author: *[email protected]，#[email protected]

Abstract Photon upconversion (UC) nanostructures providing unprecedented opportunities to manipulate the electronic excitation, energy transfer and upconverted emissions, attract extensive attentions. However, the traditional rate equations model for bulk materials fails to characterize the UC luminescence behavior of core-shell nanostructures. The understanding of core-shell upconversion emission process is not clear. Here we developed a spatial energy transfer and migration model for upconversion dynamics based on the diffusion mechanisms, which is the first demonstration to quantitatively and theoretically describe the upconversion luminescence space-time dynamics of nanostructures. This new model considering the spatial diffusion of energy migration (EM) and the coupling of the macroscopic excitation migration and the surface-related quenching on the UC luminescence, successfully overcomes the limitations of non-uniform distribution of doped ions in nanostructures on the theoretical modeling. Based on this model, two typical core-shell upconversion nanostructures (NaYF4: 20%Yb3, 2%Er3+@NaYF4 core/ inert shell; NaYF4: 20%Yb3+, 2%Er3+@NaYF4: 20%Yb3+ core/ active shell) were synthesized and excited by a 980 nm laser to experimentally determine the EM diffusion constant, the surface-related quenching parameters and the macroscopic ET rates quantitatively revealing the space-time dynamics of UC nanostructures. It provides significant theoretical guidance for predicting and modulating the luminescence performance of UC nanostructures for applications in lighting and displays, biological imaging and solar cells.

1. Introduction Lanthanide-doped materials possess unique upconversion (UC) luminescence characteristics due to abundant ladder-like energy levels of 4f configurations of Ln3+. Unlike quantum dots and organic dyes

1,2

which are sensitive to their size and chemical surroundings, Ln3+ doped UC 1

ACS Paragon Plus Environment

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materials have excellent optical stability, long lifetimes, large Stokes/Anti-stokes shifts and sharp-band emissions generation 5-8,

3,4.

Compared with two-photon fluorescence and second harmonic

the UC luminescence produced by Ln3+ doped materials requires more than 5 orders

of magnitude lower excitation powers but yield more than 5 orders of magnitude higher quantum efficiencies. As a consequence, Ln3+ doped UC materials have attracted significant attentions and found numerous applications in the fields of lighting and displays

9-11,

bio-markers

12-15,

drug-carriers 16,17, and photo-voltaic devices 18-20. Recent advances on upconversion applications are focused more on photon UC core-shell nanostructures

21-25

due to the advantage of nanophotonic control of the excitation dynamics.

Precisely engineering the material structure on the nanoscale can yield the unprecedented, effective ability to manipulate the UC behaviors including the spectrum, intensity, efficiency, lifetime and other energy transfer pathways. Capobianco et al. 26 first experimentally discovered a significant increase in the upconversion intensity of core/ active shell nanostructure compared to those of core/ inert shell, or core only nanostructures, and attributed the increased intensity to the interactions between the shell and the core. Qiu et al.

27

reported the UC luminescence

enhancement (~240 times) in a core/ active shell/ inert shell nanostructure due to the directed energy migration in the second shell layer. Zuo et al. 28 designed a core/ active shell/ active shell nanostructure to illustrate the dependence of the UC luminescence on the second shell thickness and the doping concentration of Yb3+. Monte-Carlo simulations were used to analyze the interactions between Ln3+ at the simplified conditions and explain the UC luminescence behaviors of the core/shell/shell nanostructure under the influence of excitation migration (Yb3+→Yb3+). Precise simulations require accurate microscopic rate parameters and complex coupling of the energy transitions and transfers among multiple levels, which limits its usefulness in the UC core-shell nanostructures. Despite existing demonstration on the UC photon-physical processes by experiments and simplified simulations, quantitative understandings of the UC luminescence dynamics in core-shell nanostructures are still lacking. The traditional Grant’s rate equation model 29

for UC bulk materials fails to describe the UC luminescence behavior of core-shell

nanostructures due to its characteristics: (1) The uniform spatial distribution assumption of the doped Ln3+ is not always appropriate in core-shell nanostructures. (2) The infinitely fast migration assumption is probably invalid in core-shell nanostructures. (3) The influence of the surface-related quenching mechanisms

30

on the UC luminescence dynamics need to be

considered. Thus, theoretical and quantitative illustrations of the UC luminescence of core-shell nanostructures are still quite difficult and necessary for UC luminescence modulation on nanometer scale. In this paper, to overcome these limitations, we therefore propose a new spatial energy transfer and migration model based on the diffusion mechanisms for the first time to quantitatively reveal the temporal and spatial UC luminescence behavior of core-shell nanostructures. The EM diffusion constant, the surface-related quenching parameters and the macroscopic ET rates of core-shell nanostructures are experimentally determined using this model. This analysis provides a pathway to understand and modulate the UC luminescence of core-shell nanostructures. 2


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2. Theoretical Background A. Energy Transfer and Energy Migration. In the typical sensitizer–activator (S-A) ions co-doping UC luminescence system, the S ions by absorbing the incident photons are excited from their ground state to the excited state, and then nonradiatively donate their absorbed energy to the other S or A ions via resonant interactions corresponding to energy migration (EM) or energy transfer (ET). The excited A ions deactivate from higher energy levels to produce upconverted photons. Cross Relaxations between A-A pairs

31

and cooperative Energy Transfers between

S-S-A pairs 32-34 may also exist in UC systems. According to Forster-Dexter theory 35-36, the energy transfer is through dipole-dipole interactions and the resonant energy transfer rate wSA between S-A pairs is given by ( wSS between S-S pairs also takes this form): wSA 

wS 3 hc 4 f Sems ( E ) f Aabs ( E ) ( ) Q dE A R 6 4 n E4

(1)

where wS is the radiative relaxation rate of the S ions when A ions are absent, h is the reduced Plank’s constant, c is the speed of light, n is the refractive index of the host medium, QA is the integrated absorption cross section of the activators, and the integral represents the spectral overlap between the sensitizer emissions and the activator adsorptions as a function of the photon energy E . wSA and wSS are proportional to the inverse power of the distance; therefore the ET and EM processes happen most probably between adjacent S-A and S-S pairs. Frequently there exists an energy mismatch between the energy levels of the S and A ions which is compensated by multiple phonons by the host lattice. Therefore, for lanthanide ions with similar average S-S and S-A spacing distances, wSS

is usually two or more orders of magnitude larger than wSA

indicating a much faster EM compared with ET. Only a minority of the sensitizers is capable of transferring the excitation energy to nearby activators and directly participating in the upconversion. Other isolated sensitizers participate in the upconversion processes indirectly through migration of their excitation energy to the few sensitizers adjacent to activators. ET is EM-assisted and ET processes directly produce upconversion photons while EM processes are essential for the system to maintain efficient and persistent UC luminescence. B. Grant’s Rate Equations. Grant used the first principles to develop the classical rate equations model based on the statistical analyses for the transition rates to describe the luminescence dynamics 29.The uniform distribution characteristics of S and A ions in a typical UC system ensures that the macroscopic statistical form of W representing the average transfer rate among the assembly of S and A ions can be deduced on the basis of considering all pairwise interactions between either S-S pairs or S-A pairs

29.

When wSS  wSA , the infinitely fast

migration assumption is satisfied leading to a “sea” of excitations in a metastable excited state in which the non-uniformity of the excited S ion distribution is smeared out over the excitation 3



volume. Under the assumptions of the uniform spatial distribution of the doped Ln3+ and the infinitely fast migration, the population density change rates of the energy levels characterizing the UC luminescence intensity changes from which distinct forms of the macroscopic rate equations model can be built by considering different levels and interactions of the doped Ln3+ 37-42.

3. Description of the Model A. Short-ranged and Long-ranged Characteristics of EM. Excitation migration among S ions accounts for the energy exchange between adjacent units in the UC materials. The excited S ions sustain uniform spatial distributions during ET process in bulk materials (Fig.1-(a)) so that there is no net excitation energy flux between adjacent units. Therefore, EM exhibits the short-ranged characteristics in the units of bulk materials.

Fig. 1. Schematic diagram of the divided volume units in (a) the bulk material system and (b) the core/ active shell nanostructure (The left unit is in the core and the right one is in the shell). The dashed line/curve refer to unit interface. The arrows are either ET or EM processes wherein the dashed ones refer to processes already happened while the solid ones represent processes that may happen. In (a) the non-existent net excitation energy flux between adjacent volume units emphasizes the short-ranged characteristics of EM. In (b) the net excitation migration flux between adjacent volume units driven by the non-uniform distribution of excited S ions emphasizes the long-ranged characteristics of EM.

The core-shell nanostructure is generally designed by incorporating a non-uniform spatial distribution of the doped Ln3+. Here, a core-shell nanostructure which has an active core of S and A ions and an active shell of S ions is considered as shown in Fig. 1-(b). Fig. 2 gives the schematic representation of the UC process in this specific core/ active shell nanostructure where Yb3+ and Er3+ ions separately act as sensitizer and activator ions. As illustrated in Fig. 1-(b), 4


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although the S ions are uniformly distributed, the excited S ions in the shell unit have no access to A ions at all which leads to the non-uniform distribution of excited S ions under the ET effect. The long-ranged EM is across units in the core-shell nanostructure and there inevitably exists the net excitation energy flux between adjacent units. In addition, surface-related quenching in the nanostructure is also an important cause of the non-uniform distribution of excited S ions between the internal and boundary units. The net excitation energy exchange between units in the core-shell nanostructure is attributed to the gradient dependent diffusion behavior 43-45 arising from the population density spatial gradient of excited S ions.

Fig. 2. Schematic representation of the energy level diagram illustrating the essential interaction steps in the core /active shell nanostructure, which has an active core of S and A ions and an active shell having only S ions under 980 nm excitation. Here Yb3+ and Er3+ separately act as sensitizer and activator ions. Some of the levels also exhibit multiphonon relaxations, nonradiative decays, forward or backward energy transfers, which are not shown in the diagram.

B. The Spatial Energy Transfer and Migration Equations. By introducing the diffusion constant k ss and the surface relaxation rate, we develop our model for the ions population density based on the gradient dependent diffusion mechanism and surface-related quenching effect. The change rate of the excited ions population density having both spatial and temporal characteristics in the core-shell nanostructure are given by:  2  N S 2 (r , t )  q N S 1  WS 2 N S 2  k SS  N S 2   WS 2  Ay N S 2 N Ay   WAy  S 1 N Ay N S 1 ; for core y(1...m ) y(1...m )    2  N S 2 (r , t )  q N S 1  WS 2 N S 2  k SS  N S 2 ; for shell   2 (2)  N S 2 (r , t )  q N S 1  WS 2 b N S 2  k SS  N S 2 ; for shell surface   N Ay (r , t )  WAy N Ay   y , 2  y  m; for core     N Sx (r , t )  0,  N Ay (r , t )  0   x(1,2) y(1...m )  5



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Where N Sx is the population density of S ions in the ground energy level ( x  1 ) or in the excited energy level ( x  2 ), N Ay is the population density of A ions in the ground energy level ( y  1 ) or in other excited energy levels ( y  2...m ), q is the excitation power density,  is the absorption cross section, r and t

are the space and time coordinates, WS 2 and WAy are the total

relaxation rates of excited S and A ions including both radiative and non-radiative pathways, WS 2- Ay and WAy  S1 are the forward and backward macroscopic energy transfer rates, and

y

represents the population change rate of all levels of A ions caused by all possible processes. k SS  2 N S 2 reflects the long-range effects of excitation migration between adjacent units.  2 is

the Laplacian operator of space coordinates. WS2-b is the surface relaxation rate depending on the surface-related quenching mechanisms and the surface deactivations including defects, ligands and solvents. The long-ranged excitation migration term, the surface quenching term, and the macroscopic energy transfer terms are combined together in the new model to characterize the spatial and temporal UC luminescence behavior of the core-shell nanostructure. This model having the spatial coordinates is an important improvement over the classical Grant’s rate equations. It can describe the spatial progress of EM in the core-shell nanostructure which is only indirectly observed in previous reported experiments. Based on the new model, the macroscopic rate parameters, the diffusion constant and the surfacing quenching parameters of core-shell nanostructure may be obtained by the luminescence decay experiments. On the other hand, the UC luminescence dynamics of the nanostructure can be also quantitatively analyzed using the new model and the determined parameters.

4. Comparison with Experimental Data In this section, we present luminescence steady curves and decay curves measured in NaYF4: Yb3+, Er3+ core-shell nanoparticles at different pulse excitation powers. The proposed model will be applied to properly fit all luminescence decay curves to determine the macroscopic energy transfer rates, diffusion constant and surface quenching parameters of core-shell nanoparticles. A. Experimental Methods. Two UC nanoparticles, CS-I (NaYF4: 20%Yb3, 2%Er3+@NaYF4 core/ inert shell) and CS-II (NaYF4: 20%Yb3+, 2%Er3+@NaYF4: 20%Yb3+ core/ active shell), were synthesized using the co-precipitation method and then dispersed in cyclohexane solvents at a concentration of 2 mg/ml. The transmission electron microscopy (TEM) image in Fig. 4-(a) suggests the hexagonal crystalline phase of CS-I, and the particle size distribution histograms in Fig. 3 explicitly indicates the growth of the inert/ active shell on the core. The equivalent diameter of the core is 60 nm. The inert shell thickness in CS-I is above 10 nm and the average active shell thickness in CS-II is calculated as 6.55 nm. The luminescence properties of both core-shell nanostructure samples were studied at room temperature and in the colloidal state.

6


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Fig. 3. The distribution histograms of the equivalent diameter of the top/bottom surface of (a) the core (without the shell), (b) CS-I, (c) CS-II and the distribution histograms of the height of (d) the core, and (e) CS-II. Histograms of the particle sizes are drawn from analysis of >100 particles for each sample. The mean and standard deviation for each nanocrystalline size are, respectively, 59.58±3.85 nm in (a);

82.75±3.55 nm in (b);

68.23±2.58 nm in

(c); 45.40±3.21 nm in (d); 61.41±3.21 nm in (e).

A 980 nm CW diode laser (MDL-H-980-5W) and a pulse signal generator (DG1022Z, RIGOL) were used to generate pulsed 980 nm excitation with tunable frequencies and duty ratios in luminescence experiments.The steady-state spectra of the UC nanoparticles were measured using the Andor shamrock SR-500i imaging spectrometer. The laser spot area is about 48 mm2. The luminescence decay curves of the UC nanoparticles were measured using a customized phosphorescence lifetime spectrometer (FLS 980, Edinburgh) equipped with a single emission monochromator and a UV/VIS or NIR PMT together covering the wavelength range from 200 nm~1700 nm. The pulse laser beam was focused onto the UC samples loaded in a quartz cuvette with a path length of 1 cm. The UC emissions were collected at an angle of 90° to the excitation beam by a pair of lenses.The peak power density of the incident 980 nm pulse laser was measured to be about 11.1W/cm2 with the laser spot area of 8 mm2. B. Power Dependence (Steady State). To clarify the detailed interactions among levels, a 980 nm CW laser was used as the excitation light to experimentally obtain the steady-state upconversion emission spectra of core/ inert shell nanostructure CS-I at various excitation powers 7



(the power density below 6W/cm2) which are shown in Fig. 4. All significant energy transitions, transfers and migrations between 2 levels of Yb3+ and 9 levels of Er3+ 46 were considered in Fig. 2. Red (4F9/2→4I15/2), green (2H11/2, 4S3/2→4I15/2) and blue (2H9/2→4I15/2) upconversion emissions are observed in Fig. 4. For multi-step ET processes, the luminescence intensity I is well known to be proportional to the n power of the excitation power P 47, 48, so the number of energy transfer steps has an essential link to the log-log plot of I versus P. A two-photon process contributes to the green emission which arises from the 2H11/2, 4S3/2 levels with sequential energy transfers from Yb3+ wherein the Er3+ is excited first from 4I15/2 to 4I11/2, and then from 4I11/2 to 4F7/2, which subsequently nonradiatively relaxes to the emitting 2H11/2, 4S3/2 manifold. The slopes of 1.44 and 1.47 indicates that the red (654 nm) and blue (410 nm) upconversion emissions most probably come from similar processes: a) The blue emission arises from the 2H9/2 level via a three photon process involving sequential energy transfer where the Er3+ is promoted to or just above the blue-emitting 2H9/2 from the green-emitting 2H11/2, 4S3/2. b) A most probable contribution of the red emission is from a three-photon process corresponding to a backward energy transfer which deactivates the Er3+ from the higher 4G, 2K, to 4F9/2 rather than a two-photon process which excites the Er3+ from the lower 4I13/2 to 4F9/2.

Fig. 4. (a) Transmission electron microscopy characterization of the β-NaYF4: 20% Yb3+, 2% Er3+@NaYF4 core/ inert shell upconversion nanoparticles (UCNPs) indicating morphology. Scale bars are 100 nm. (b) Steady-state upconversion emission spectra of core/ inert shell nanostructure CS-I under 980 nm continuous excitation at various powers (the power density below 6W/cm2). Clear Red (654 nm), green (540 nm) and blue (410 nm) upconversion peaks were observed with the emission intensities increasing as the excitation power rises.

Thus, the complete form of Eq. (2) reflect the space-time dependent population changes of all 

the Yb3+ and Er3+ energy levels wherein the specific forms of N Ay (r , t ) terms are given by:

8


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

N A9  WNR 9 N A9  (WS 2  A6 N S 2 N A6  W A9  S1N S1N A9 ) 

N A8  (WNR8  WR8 ) N A8  WNR 9 N A9  (WS 2  A5 N S 2 N A5  W A9  S1N S1N A9 ) 

N A7  WNR 7 N A7  WNR8 N A8  (WS 2  A3 N S 2 N A3  W A7  S1N S1N A7 ) 

N A6  (WNR 6  WR 6 ) N A6  (WNR 7 N A7  WS 2  A6 N S 2 N A6  WCR 6 N A1N A6 )

(3)



N A5  (WNR 5  WR 5 ) N A5  WNR 6 N A6  85WR8 N8  (WS 2  A2 N S 2 N A2  WS 2  A5 N S 2 N A5  W A9  S1N S1N A9 ) 

N A4  WNR 4 N A4  WNR 5 N A5  (WUC 2 N A2 N A2  WCR 4 N A1N A4 ) 

N A3  (WNR 3  WR 3 ) N A3  WNR 4 N A4  83WR8 N A8   63WR 6 N A6  53WR 5 N A5  (WS 2  A1N S 2 N A1  WA7  S1N A7 N S1  WA3 S1N A3 N S1  WS 2  A3 N A3 N S 2  WCR 6 N A1N A6 ) 

N A2  WR 2 N A2  WNR 3 N A3  82WR8 N A8   62WR 6 N A6  52WR 5 N A5  32WR 3 N A3  (WCR 6 N A1N A6  2WCR 4 N A1N A4  2WUC 2 N A2 N A2  WS 2  A2 N S 2 N A2 )

Where WNRy and WRy represent the non-radiative and radiative relaxation rates of excited Er3+, WCRy is the cross-relaxation rate between two corresponding energy levels of excited Er3+, WUC 2

is the additional energy transfer rate between two excited Er3+. ij were previously determined from emission spectra and calculated from Judd-Ofelt parameters, which are assumed known parameters here as the branching ratios determined by Feng et al 49. C. Experimental and Simulated Luminescence Decay Curves of CS-I. The inert shell in the CS-I is reasonably assumed thick enough to prevent surface quenching of Er3+ and Yb3+ in the core. The UC luminescence intensities of CS-I are dependent on WS 2 , WNRy , WRy , WS 2  Ay , and WAy  S 1 . The luminescence decay simulations of CS-I were carried out using Eqs. (2) and (3) by

the discretization of time coordinates. The EM term and the surface quenching term in the equations may be neglected. The experimental and simulated UC and DC emission decay curves of CS-I (Fig. 5-(a)) following 980 nm pulsed excitation at various pulse conditions are shown in Fig. 5-(b)~(d). The simulated population density changes were fit to the experimental curves to determine 12 relaxation rates ( WNRy , WRy and WS 2 of excited Er3+ and Yb3+) and 10 macroscopic energy transfer rates ( WS 2 Ay , WAy  S1 between Yb3+-Er3+ pairs and WCRy , WUC 2 between Er3+-Er3+ pairs) which were listed in Table 1. Standard Parameter

Averaged values

Standard Parameter

deviation

Radiative Decay Rates (s-1)

Averaged values

deviation

Nonradiative Relaxation Rates (s-1)

WS2

936

± 33

WNR9

2.96×106

± 0.03×106

WR8 (blue)

1165

± 97

WNR8

2.36×104

± 0.09×104

9



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WR6 (green)

1773

± 150

WNR7

1.42×106

± 0.28×106

WR5 (red)

2118

± 52

WNR6

52.3

± 24.1

WR3 (1 μm)

66

± 13

WNR5

-

-

WR2 (1.5 μm)

106

±4

WNR4

2.34×104

± 0.05×104

73.2

± 18.2

WNR3 Macroscopic ET rates between

Yb3+-Er3+

pairs and

Er3+-Er3+

pairs

(cm3s-1)

WS2-A1

1.42×10-15

± 0.32×10-15

WA7-S1

1.90×10-16

± 0.31×10-16

WS2-A2

-

-

WA9-S1

1.02×10-16

± 0.02×10-16

WS2-A3

1.32×10-15

± 0.25×10-15

WCR6

1.86×10-17

± 0.19×10-17

WS2-A5

1.65×10-15

± 0.28×10-15

WCR4

5.02×10-19

± 2.42×10-19

WS2-A6

7.60×10-15

± 1.96×10-15

WUC2

2.27×10-17

± 0.41×10-17

WA3-S1

2.14×10-16

± 0.46×10-16

Table.1. Eq. (2) were solved using the Euler method with a time step of 0.5 μs. The time-dependent simulated population densities of excited Er3+ were compared with the shapes of experimental upconversion emission decay curves based on the least square method. From 10 different fits with randomly chosen starting points, the averaged best-fitting macroscopic parameters and their standard deviations determined from the core/ inert shell nanostructure CS-I are listed above.

Fig. 5. (a) Schematic illustration of the UC process in the core/ inert shell nanostructure CS-I under NIR excitation. Following are the experimental and simulated UC and DC emission decay curves of CS-I under 10


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different pulsed excitations at 980 nm of (b) 10 ms pulse cycle and 0.3 ms pulse duration. (c) 10 ms pulse cycle and 1.0 ms pulse duration. (d) 100 ms pulse cycle and 5.0 ms pulse duration (1.5 μm DC emission cannot reach a complete relaxation state in 10 ms so a larger pulse cycle was chosen). The colored dotted curves represent experimental data while black solid curves represent the simulations. All the curves were normalized to the same maximum to compare the shapes as opposed to the brightness and then to different maximums to achieve a clear display effect.

D. Experimental and Simulated Luminescence Decay Curves of CS-II. The active shell in the CS-II is thick enough to prevent surface quenching of Er3+ in the core while still have surface quenching of the large amount of Yb3+ in the shell. Therefore the UC luminescence intensities of CS-II will be greatly influenced by the diffusion constant kss , the surface quenching rate WS 2b and the effective thickness d of the surface quenching region. In the simulations, the reasonable simplifications of nanoparticle geometry were taken to simulate the luminescence dynamics of the CS-II using Eq. (2). Statistically a large amount of hexagonal-prism-shaped particles with slightly different sizes can be assumed spherical with the same averaged equivalent volume regarding both the core and the core/ shell structure. Then the averaged core diameter and active shell thickness are calculated to be respectively 62.3 nm and 6.55 nm. This treatment puts the spatial discretization process in the sphere coordinates and reduces its complexity wherein the form of the spatial gradient  2 N in the sphere coordinates is easily derived with the central difference method. In addition, the size of the divided volume mesh units in each nanoparticle is assumed around 1~3 nm which is not too large to guarantee the validity of short-ranged EM inside the mesh unit. It is also worth noting that our model is a statistical macroscopic model characterizing the population density variations, so that the volume unit can be relatively small even if not including many lanthanide ions. The experimental and simulated UC emission decay curves of CS-II (Fig. 6-(a)) following 980 nm pulsed excitation at various pulse conditions which are compared with the experimental decay curves of CS-I are shown in Fig. 6-(b)~(d). The UC red, green and blue emission lifetimes of CS-II in experiments are obviously shorter than those of CS-I due to the coupling effect of the energy migration and the surface-related quenching. If without surface-related quenching, the excitation energy migrates among the Yb3+ from the shell to the core which will brighten the emissions and prolong the lifetimes. Therefore, the shortening lifetime of CS-II verifies the occurrence of surface-related quenching of Yb3+.

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Fig. 6. (a) Schematic illustration of the UC process in the core/ active shell nanostructure CS-II under NIR excitation. Following are the experimental and simulated 410 nm, 540 nm and 654 nm UC emission decay curves of CS-II compared with those of CS-I under different pulsed excitations at 980 nm of (b) 10 ms pulse cycle and 0.3 ms pulse duration. (c) 10 ms pulse cycle and 1.0 ms pulse duration. (d) 100 ms pulse cycle and 5.0 ms pulse duration. The colored solid curves represent the experimental data of CS-I and CS-II while black solid curves represent the simulations of CS-II.

With the energy migration and the surface-related quenching mechanisms included in the modelling of CS-II after predetermining 22 rate parameters from CS-I, the experimental data were fit to the simulated curves to get a diffusion constant of kss = 2.13×10-12 m2/s, a surface quenching shell thickness of d = 0.92 nm, and a surface quenching rate of WS2-b =9.2×103 s-1 (considerably larger than the relaxation rate at the internal regions). The results in Fig. 6 shows that the simulation decay curves and the experimental curves agree well to guarantee the solution accuracy of (kss, d and WS2-b). kss may be regarded as a measure of the excitation energy migrates distance (kss t )1/ 2 in a time t 43 so that the measured diffusion constant kss = 2.13×10-12 m2/s corresponds to a

migration distance of 46 nm in 1 millisecond (the scale of the luminescence lifetimes). This migration distance is much larger than the unit size in the core-shell nanostructure which ensures the assumption of infinitely fast migration in the unit and close to the nanostructure size indicating that the long-ranged excitation migration between units cannot be treated as infinitely fast. The EM diffusion constant is essential for characterizing how fast the excitation migrates among the sensitizer ions. This verifies the necessity and applicability of introducing the gradient dependent 12


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diffusion term in the model.

5. Space-Time Dynamics Characterization To gain further insight about the coupling effect of surface-related quenching and energy migration on the UC dynamics of the core/ active shell CS-II, Fig. 7 shows the simulated 540 nm green emission decay curves for various parameters based on Eqs. (2) and (3). The simulated 540 nm emission lifetime can be tuned from 720 μs to 500 μs by varying the quenching rate at the surface from 1000 s-1 to 10000 s-1. Since the near-surface excited Yb3+ ions have a faster quenching rate, the monotonous decreasing trend of 540 nm emission lifetime can be explained by the weakened forward energy transfers from Yb3+ to Er3+ together with the enhanced backward energy transfers from Er3+ to Yb3+. When the diffusion constant increases from 1×10-13 m2/s (relatively slow) to 5×10-12 m2/s (very fast), the 540 nm emission lifetime also exhibits a monotonic decreasing trend but levels off with faster migration. This phenomenon reveals that: a) The influence of the active shell on the core appears insignificant when EM is relatively slow. With increased diffusion constant, this influence becomes significant. b) There exists a critical value of the diffusion constant above which the EM are deemed infinitely fast and the emission lifetime no longer varies. It is noted that the surface-related quenching rate as well as the diffusion constant may be adjusted by either designing the shell or modulating the ions doping concentration, which reflects the practical significance of model and simulation.

Fig. 7. Simulated UC emission decay curves of the core/ active shell nanostructure CS-II when the parameters of (a) surface-related quenching parameter WS2-b (b) diffusion constant kss varies.

Figure 8 shows the calculated results of the temporal and spatial population density distributions of excited Er3+(4S3/2) and Yb3+(2F5/2) in CS-II under 980 nm pulsed excitation. The temporal behaviors are obtained as follows: (1) The population densities of excited Er3+(4S3/2) in the core and Yb3+(2F5/2) in the core and shell increase with the experimental time during the excitation before 300 μs. (2) Short after stopping the 980 nm excitation at 300 μs , an immediate decrease of the excited Yb3+(2F5/2) population density in both core and shell is observed due to the lack of the excitation source, while a continued increase of excited Er3+(4S3/2) population density in the core is also observed. This is because the continuous energy transfer (Yb3+→Er3+) after 13



stopping the excitation source still promote the UC emission and the lagged luminescence effect will last for a short period of time. (3) Long after stopping the excitation (>500 μs), the population densities of excited Er3+(4S3/2) and Yb3+(2F5/2) decreases prominently. The simulation results revealed the spatial behaviors of the excited population densities relating to UC emission of core-shell nanostructure, which have never been qualitatively considered in existing research. The population density spatial distribution has different characteristics at different experimental time. Some typical time points (100 μs, 300 μs, 500 μs and 700 μs) are chosen here for analyses: (1) At t = 100 μs (during the excitation), a monotonic increase of excited Er3+(4S3/2) population density in the core along x1 is observed implying the actual from-shell-to-core effect of the excitation migration. The excited Yb3+(2F5/2) population density first increases and then decreases along x2 due to the surface-related quenching. (2) At t = 300 μs ( the time of stopping the excitation), similar spatial distribution characteristics of excited Er3+(4S3/2) and Yb3+(2F5/2) population densities are observed while the trends become more obvious. (3) At t = 500 μs (the initial stage of stopping the excitation), the excited Er3+(4S3/2) population density starts to show a decreasing trend along x1 though not obvious while the excited Yb3+(2F5/2) population density clearly decreases along x1. This can only be explained by the from-core-to-shell excitation migration. Since Yb3+(2F5/2) ions quench much faster at the surface, the area acts as a “sink” causing the reverse of excitation migration direction. This phenomenon of the “reverse” is also verified by the inset of contour plot in (b). The spatial gradient of excited Yb3+(2F5/2) population density at the core-shell interface (x2=0) clearly illustrates that the excitation migrates from the shell to the core at first (positive dNS2/dx2) and later takes the opposite direction (negative dNS2/dx2). (4) At t = 700 μs (Long after stopping the excitation), the decreasing trends of the excited population densities appears more obvious. Above all, we firstly explicitly demonstrate here that EM among S ions possess long-ranged characteristics in core-shell nanostructures and the excitation migration may take opposite directions as time goes by, which will also change previous understanding of UC core-shell nanostructures.

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Fig. 8. (a) and (f) show the contour plots of the temporal and spatial population density distributions of excited Er3+(4S3/2) in the core which produce 540 nm emission and excited Yb3+(2F5/2) in both core and shell which produces 1 μm emission. (b)~(e) and (g)~(j) respectively show the spatial population density distribution curves of Er3+(4S3/2) and Yb3+(2F5/2) at several time points (100 μs ~700 μs). The core/ active shell nanostructure is under 980 nm pulsed excitation (10 ms pulse cycle and 300 μs pulse duration). Coordinate x1 in (a) represents the distance to the sphere center. Coordinate x2 in (f) represents the distance to the core-shell interface. Inset of contour plot of (f) is the time evolution of the spatial gradient of excited Yb3+(2F5/2), dNS2/dx2, at the core-shell interface (x2=0).

6. Conclusions A spatial energy transfer and migration model based on the diffusion mechanisms was developed to accurately characterize the UC luminescence of the core-shell nanostructure having non-uniformly distributed doped ions. The long-ranged excitation migration term, the surface quenching term, and the macroscopic energy transfer terms are combined together in this model. Excitation migration and surface-related quenching both greatly affect the UC behavior of the nanostructure. The excitation migration balances the uneven distribution of the excited Yb3+ in different locations in the nanostructure, while the surface-related quenching causes the excited Yb3+ at the shell surface to relax much faster. The model is based on an improved understanding of the mechanisms and careful observations for the luminescence to overcome the limits of experimental trials and predict the spatial and temporal luminescence behavior of the core-shell nanostructure. The simulations based on this model are in good agreement with the experimental curves for NaYF4: Yb3+, Er3+ core-shell nanoparticles to verify this model. This analysis provides significant theoretical guidance for predicting and modulating the luminescence performance of UC nanostructures in applications.

Author information Corresponding Author

*E-mail: [email protected]，[email protected] 15



Notes

The authors declare no competing financial interest.

Acknowledgments This work was supported by the National Key Research and Development Program of China (No. 2016YFC0802500), the National Natural Science Foundation of China (Nos. 51576110, 51622403) and the Science Fund for Creative Research (No. 51621062). The authors have no competing interests to declare. We thank Prof. D.M. Christopher for editing the English.

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