The molecular weight of NaYF4:RE photonic up-conversion nanoparticles

10 Luminescence up-conversion nanoparticles (UCNPs) consisting of a NaYF4 crystal lattice doped with rare earth (RE) ions have found widespread application in bio-sensing, bio-imaging, and therapeutics; yet the molecular weight of UCNPs is not known. Lack of knowledge of molecular weight of UCNPs results in sub-optimal functionalisation and dosages of UCNPs. We present a simple method for calculating the molecular weight of NaYF4:RE UCNPs from arbitrary crystal lattice parameters and UCNP diameter measurements, and we apply this method to estimate the molecular weight of various NaYF4:RE UCNPs from the literature. UCNP molecular weight scales exponentially with UCNP volume (i.e. diameter cubed). UCNPs of 10 nm diameter are estimated to be a molecular weight of ~ 1 MDa, and 45 nm UCNPs are estimated to be ~100 MDa. Hexagonal lattice UCNPs were found to have a greater molecular weight than their cubic lattice UCNP counterparts. A Gaussian distribution 20 of nanoparticle diameters was found to produce a lognormal distribution of nanoparticle molecular weights. We provide stand-alone graphic user interfaces to calculate UCNP:RE molecular weight. This approach can be generalised to estimate the molecular weight of crystalline nanoparticles of arbitrary size, geometry, and elemental composition where nanoparticle unit cell parameters are known.


Introduction
Up-converting nanoparticles (UCNPs) consisting of a low phonon-energy crystal lattice host (typically NaYF4) co-doped with rare earth (RE) ions (typically Yb 3+ with Er 3+ and/or Gd 3+ ) have recently generated widespread scientific interest as optical platforms for numerous applications, including: 30 imaging contrast labels in cells and animals in vivo and ex vivo 1-10 ; the detection of antibiotics 11 or toxins in food [12][13][14] ; aggressive targeting of cancer cells 15,16 ; imaging thermometry 17,18 ; enhancement of photovoltaic technology, 19,20 and the measurement of biomarker molecules in biological fluids. [21][22][23][24][25][26] NaYF4:RE UCNPs have garnered intense interest due to their unique optical luminescence excitation and emission properties, which are highly advantageous for bio-sensing and bio-imaging applications. NaYF4:RE UCNPs absorb multiple near infra-red (NIR) photons (typically ~ 980 nm) and emit luminesce at visible wavelengths via photonic up-conversion. 27 NIR excitation is highly advantageous for biomedical applications because: (1) NIR light can penetrate several centimetres in tissue 28 ; (2) NIR light is not phototoxic, and (3) NIR excitation results in no auto-fluorescence from 40 tissue or endogenous proteins. The wavelength and intensity of UCNP luminescence emission can be tuned by altering UCNP composition, 29 the concentrations of RE dopants, 30 altering the UCNP crystal lattice phase, 5,29 and altering nanoparticle surface area to volume ratio. 31 It has become common practice to add protective inert outer shells -typically made of silica -to shield UCNPs from quenching effects by solution and enhance luminescence emission intensity. 32,33 Thus, the optical properties of UCNPs can be tailored to suit individual applications. The surface of NaYF4:RE UCNPs are typically functionalised with antibodies 22,[24][25][26] or oligonucleotides 34 to enable specific binding for labelling, bio-sensing, and therapeutic applications. Close proximity of molecules (e.g. a target analyte protein) to the surface of UCNPs results in luminescence energy resonance transfer (LRET) between the UCNP (donor) and a proximal molecule (acceptor), resulting in altered UCNP 50 luminescence emission. The resulting change in luminescence emission depends on the particular physical configuration and optical properties of the UCNP and acceptor molecule. 2,[24][25][26]35,36 Despite the widespread applications of NaYF4:RE UCNPs, the molecular weight of these nanoparticles has not been reported. Consequently, the concentration of UCNPs is typically reported in terms of UCNP weight per volume of aqueous media. 37 This relatively crude measure does not allow the number of UCNPs in a given sample to be known and consequently may result in suboptimal functionalisation of UCNPs or inaccurate estimation of UCNP dosage for bio-imaging and therapeutic applications. Further, knowing the molecular weight of UCNPs would be helpful in understanding UCNP behaviour when trapped with optical tweezers 38 and understanding the uptake of UCNPs by cell membranes. 3,5,39 60 This study presents the theory required to calculate the molecular weight of NaYF4:RE UCNPs of arbitrary size and RE dopant composition, assuming that crystal lattice parameters are known. The molecular weight of NaYF4:RE UCNPs can be calculated from first principles if the lattice parameters and elemental composition of NaYF4 crystal unit cells are known from X-ray diffraction (XRD) experiments; and if the geometry and diameter of UCNPs are known from transmission electron microscopy (TEM) experiments. 5,6,40 The molecular weight of a UCNP can be estimated by simply dividing the volume of a UCNP by the volume of an individual unit cell to get the number of unit cells in a UCNP. From this, the total molecular weight of a UCNP is calculated summing the atomic weight of all atoms within a single unit cell, and multiplying by the total number of unit cells within the UCNP. From this theory, we estimate the molecular weight of various NaYF4:RE UCNPs reported in 70 the literature. Of particular note, we show that a Gaussian distribution of NaYF4:RE UCNP diameters will result in a lognormal (i.e. non-normal) distribution of NaYF4:RE UCNP molecular weights.
Additionally, we provide two stand-alone graphical user interfaces (GUIs) to enable others to easily calculate the molecular weight of NaYF4:RE UCNPs with arbitrary size, lattice, and RE dopant parameters.

Structure and photonic up-conversion of NaYF4:RE UCNPs
In NaYF4:RE UCNPs, photonic up-conversion is enabled by the sequential absorption of two or more near-infrared photons, which, via excitation of several long-lived metastable electron states, and subsequent non-radiative multi-phonon and radiative relaxation, produces luminescence 80 emission at visible wavelengths. Up-conversion requires a crystalline host lattice, which is doped with multiple different lanthanide ions (typically Yb 3+ and Er 3+ ), where one lanthanide ion (typically Yb 3+ ) acts as a photo-sensitizer and the other (typically Er 3+ ) emits photons. 33 The up-conversion process is sensitive to the distance between ions and unit cell geometry. Although many different combinations of lattice and RE dopants have been explored, 41 the combination of Yb 3+ and Er 3+ ions in a NaYF4 host lattice has been found to provide high up-conversion efficiency, and as such is commonly used for UCNPs. 27,42 NaYF4:RE unit cells are typically a cubic crystal lattice arrangement or a hexagonal crystal lattice arrangement. The arrangement of unit cells influences the crystal lattice parameters, consequently changing photonic properties and density of NaYF4:RE UCNPs. Thus, to calculate the molecular 4 weight of a NaYF4:RE UCNP for a given size, the atomic weight of a single NaYF4 : RE UCNP unit cell must be determined.

NaYF4:RE unit cells reported in the literature
Several studies have reported crystal lattice parameters for cubic and hexagonal NaYF4:RE UCNPs obtained via x-ray diffraction studies (see Table 1).   40 report unit cell parameters for cubic (α phase) and hexagonal (β phase) unit NaYF4:RE unit cell configurations (see Figure 1). In the face-centred cubic lattice arrangement, high-symmetry cation sites are formed, which are randomly occupied by either Na or RE ions (see Figure 1a). Cubic unit cells follow the formula Na2Y2F8, with Y ions substituted for other RE ions depending on doping parameters. Hexagonal unit cells follow the formula Na1.5Y1.5F6. 40 In the hexagonal lattice arrangement, there are two low-cation 100 symmetry sites, which contain either Na or RE ions (see Figure 1b).

NaYF4:RE size distributions
Synthesising NaYF4:RE UCNPs typically creates spherical UCNPs with a range of diameters.

Number of NaYF4 unit cells in a UCNP
The volume of a spherical nanoparticle ( ) is given by: where is the radius of the UCNP. NB: Equation 1 can be changed to calculate arbitrary non-spherical nanoparticle geometries if desired. If the cubic lattice parameter is known, then the volume of a cubic unit cell ( ) is calculated by: If the hexagonal lattice parameters ℎ and ℎ are known, then the volume of a hexagonal unit cell The number of unit cells in a given UCNP ( ℎ ) can then be calculated by:

Atomic weight of NaYF4 unit cell 130
Assuming no RE dopants, the atomic weight of a single cubic NaYF4 ( ) or hexagonal NaYF4 unit cell ( ℎ ) are described by: Where , , and are the atomic weight of Sodium, Yttrium, and Fluorine respectively (see Table 2). If RE dopant ions are added, a fraction of Y 3+ ions are substituted for RE 3+ dopant ions, and thus the average atomic mass of unit cells within an UCNP is altered. Thus, RE doping can be accounted for by defining an additive-factor ( ): where 1 to is the fractional percentage of up to RE dopants. The additive factor is a numeric value between 0 and 1, representing 0% and 100% Y substitution, respectively. The atomic mass of a single cubic or hexagonal unit cell whilst accounting for RE dopants can thus be calculated by: 140 Where and ℎ are the average atomic weight of RE doped cubic and hexagonal unit cells, respectively.

Molecular weight of a NaYF4 UCNP
Assuming that unit cells are distributed uniformly throughout the UCNP, the molecular weight of a cubic lattice UCNP ( ) is calculated from Equations 4 and 9 as: 150 From Equations 5, the molecular weight of a hexagonal lattice UCNP ( ℎ ) is calculated by: From Equations 4, 5, 11, and 12, it can be seen that the molecular weight of UCNPs scales with UCNP volume: i.e. diameter cubed.

Molecular weight of cubic and hexagonal NaYF4 UCNPs
Using the theory presented in Sections 2.4 -2.6, the molecular weight of NaYF4 hexagonal and cubic lattice UCNPs were calculated, assuming the following generic lattice parameters: cubic lattice parameter a = 5.51 Å; hexagonal lattice parameters a = 5.91 Å, c = 3.53 Å.

UCNP diameter distribution vs. molecular weight distribution
The size distribution of a single batch of NaYF4:YbEr UCNPs was reproduced from data presented in Sikora et al. 5 A Gaussian fit was applied to this size distribution data by using non-linear least squares fitting in MATLAB (MATLAB 2016a, MathWorks) and the corresponding molecular weight distribution was calculated by the theory presented in Sections 2.4 -2.6.

Stand-alone GUIs for calculation of nanoparticle molecular weight
Two graphic user interfaces (GUIs) were written in MATLAB, each incorporating different features. The first simple GUI was developed to enable other researchers to calculate the molecular weight of spherical NaYF4:RE UCNPs for a user-defined nanoparticle size range. The second, more 180 powerful, GUI was designed to enable users to estimate the molecular weight of nanoparticles with arbitrary nanoparticle geometry; arbitrary lattice parameters; and arbitrary elemental composition, across a user-defined range of characteristic nanoparticle sizes. Additional technical information for both GUIs is provided in the supplementary material section. The stand-alone GUIs developed are shown in supplemental Figures S1 and S2.

Molecular weight of cubic and hexagonal NaYF4:RE UCNPs
As expected, the molecular weight of UCNPs scaled proportionally to the volume of the UCNPs, resulting in a large increase of UCNP molecular weight for a comparatively small increase in UCNP 190 diameter. Hexagonal lattice UCNPs were found to have a considerably greater molecular weight than cubic lattice UCNPs due to the lower volume of hexagonal unit cells (see Figure 2).

Figure 2. UCNP diameter vs. UCNP molecular weight for hexagonal and cubic NaYF4
UCNPs. Lattice parameters are assumed to be: a = 5.51 Å for cubic UCNPs; a, c = 5.91 Å and 3.53 Å for hexagonal UCNPs. (a) The exponential increase of molecular weight with diameter is apparent. (b) The same data on a logarithmic scale.

The effect of RE doping
The effect of RE ion doping on molecular weight of UCNPs was investigated by varying the percentage of RE dopants. Increasing Yb 3+ or Er 3+ was found to increase the molecular weight of UCNPs (see Figure 3). The increased UCNP mass is because Yb 3+ and Er 3+ have a greater atomic mass than Y 3+ . The difference in molecular weight of UCNPs doped with Yb 3+ and Er 3+ was relatively small to the small difference between the atomic weight of Yb 3+ and Er 3+ (see Table 2). Hexagonal lattice UCNPs show a slightly higher increase in molecular weight for a given dopant concentration than cubic lattice UCNPs due to the smaller unit cell volume of hexagonal lattices compared to cubic lattices.

Molecular weight of NaYF4:RE UCNPs reported in the literature
The estimated molecular weight of various NaYF4:RE UCNPs calculated from values reported in 220 the literature are shown in Figure 4.

UCNP diameter distribution vs. molecular weight distribution
UCNP diameter distribution data was reproduced from Sikora et al., (2013) (see Figure 5a) and was well-fitted by a Gaussian distribution (R 2 = 0.96). The corresponding molecular weight 230 distribution was then calculated for each UCNP diameter. The resultant molecular weight is shown in Figure 5b. A Gaussian fit to the molecular weight distribution plotted on a logarithmic scale produces an excellent fit (R 2 = 0.98), establishing that a Gaussian diameter distribution of nanoparticles follows lognormal distribution of molecular weight (see Figure 5c).

Discussion
Our method to estimate the molecular weight of NaYF4:RE UCNPs relies on two basic assumptions: 1. that UCNPs are of homogenous composition, and 2. that the lattice parameters are accurate. The first assumption can be tested by TEM and XRD, and the second assumption can be experimentally verified via XRD. Ensuring accurate lattice parameters is particularly important when estimating the molecular weight of UCNPs with arbitrarily large dopant concentrations. For example,   40 experimentally demonstrated that by doping a hexagonal phase NaYF4:Yb,Er UCNP (18% Yb, 2% Er) with increasing concentrations of Gd 3+ increases the lattice parameters of the UCNP significantly, resulting in an increased unit cell volume. Thus, because of this dependence of 250 lattice parameter on RE dopant percentage, our estimations of UCNP molecular weight in Figure 3 may be an over-estimation on true values if lattice parameters are not independently verified for each RE dopant concentration of interest. We assumed a spherical geometry here, but our method could be trivially adapted for arbitrary nanoparticle geometries; e.g. rods, 40,45 triangular, 46 or prismshaped 47 nanoparticles. The extension of our technique to arbitrary geometries, arbitrary crystal lattice parameters, and arbitrary elemental composition is demonstrated by the development and application of an advanced GUI incorporating all of these variables (see Figure S2).Further, our method does not account for any surface functionalisation or enhancement. Thus the molecular weight of UCNPs modified by addition of a silica 24,32,33 or calcium fluoride 48 shell coating will be somewhat different from the molecular weight estimated by our technique. Our method could be 260 augmented by estimating the molecular weight of UCNP shell coating layers and thus provide enhanced molecular weight information for surface functionalisation applications. The method presented here could be applied to nanoparticles of varying composition (e.g. silica or gold nanoparticles) with known crystal lattice parameters, known unit cell volume, and known nanoparticle size/geometry. It should be noted that a simple theory for estimation of the molecular weight of a single homogenous gold nanoparticle was proposed by Lewis et al. (2006), 49 but this theory did not account for unit cell parameters or elemental doping. Further, it did not describe the molecular weight distribution of a population of nanoparticles. In future, it would be ideal to test the accuracy of our molecular weight estimation against an experimental method for determining molecular weight such as size exclusion chromatography, 50 mass spectrometry, 50 or sedimentation-270 velocity analytical ultracentrifugation. 51,52 It is of note that for a given population of UCNPs with a Gaussian diameter distribution the corresponding molecular weight distribution will be lognormal (see Figure 5). This lognormal distribution arises from the fact that molecular weight scales exponentially with diameter. This lognormal distribution will be important to consider, in that a large fraction of UCNPs will be of a significantly lower or higher molecular weight than the arithmetic mean UCNP.
Estimation of molecular weight of NaYF4:RE UCNPs could be of use in multiple applications.
Knowledge of UCNP molecular weight will likely be of great utility in studies where UCNP surfaces 280 are functionalised with additional molecules such as antibodies 22,[24][25][26] or oligonucleotides. 34 If the molecular weight of UCNPs is known, then the molar concentrations of substances in the functionalisation processes can be altered to reach desired parameters. When combined with estimation of UCNP surface area, this could inform the optimisation of functionalisation for a given application. Knowing the molecular weight of UCNPs would also be beneficial in the processing of particles for downstream applications. In particular, steps taken to functionalise the nanoparticles may require separation procedures to remove unreacted moieties or unwanted reactants. If the molecular weight of UCNPs were known, then it may be beneficial for the optimisation of conjugation stoichiometry, which can be concentration dependant. The reaction rates of UCNPs will be heavily influenced by their molecular weight; thus a greater understanding of their molecular weight may 290 increase the knowledge of thermodynamic properties of the UCNP System. This is particularly important when considering the use of bio-receptors with UCNPs where the mass of the particle may affect the binding kinetics of the UCNP-receptor construct.
The molecular weight of UCNPs will also be of interest in the study of cytotoxicity, bio-distribution, cellular uptake, and clearance of UCNPs in biological systems. 3,5 Currently, it is difficult to compare the results from various imaging and therapeutic studies because UCNP concentration is reported as weight of UCNPs per volume of aqueous media (i.e. mg/mL or similar). 3 This is a crude measure which does not quantify number of UCNPs in a given sample. For example, nanoparticles can induce membrane damage 39 and initiate apoptosis (programmed cell-death). 53,54 Thus, understanding the molar concentration of UCNPs would help assessment of cytotoxicity effects. A standardised 300 protocol based on molecular weight of UCNPs would help assessment of where UCNPs accumulate in vivo and what time is required to clear the UCNPs from the various organs (e.g., lung, liver, kidney, heart) 4 or tumours. 48 Reporting the molar concentration of UCNP composites may also help to develop highly-localised targeted delivery of therapeutic drugs to the required sites in the body, leading to better controlled targeted photodynamic therapy, 15 and for improved targeted drug delivery. 16

Conclusions
We have presented a theoretical method to estimate the molecular weight of hexagonal and cubic crystal lattice phase NaYF4:RE UCNPs from experimentally measured UCNP crystal lattice 310 parameters and nanoparticle diameters. Our calculations assume spherical NaYF4:RE UCNP geometry, but our technique can easily be extended to estimate the molecular weight for nanoparticles with arbitrary geometries (e.g. cubes, rods, triangles, or tetrahedrons) and with arbitrary crystal lattice (e.g. cubic, hexagonal, orthorhombic), and arbitrary elemental compositions.
It should be noted that our method does not account for any surface modification of nanoparticles, only the molecular weight of the NaYF4:RE lattice itself. We expect this method will be highly accurate in estimation of the molecular weight of NaYF4:RE UCNPs where the crystal lattice parameters of non-surface modified UCNPs have been experimentally verified. Direct experimental validation of our technique is desirable. 320 UCNP molecular weight was found to increase exponentially with UCNP diameter. Thus a small increase in UCNP diameter results in a large increase in UCNP molecular weight. For example, the molecular weight of NaYF4 UCNPs were estimated to be ~ 1 MDa for a UCNP of 10 nm diameter, and ~ 100 MDa for a UCNP of 45 nm diameter. From this relation, we demonstrated that a population of UCNPs with a Gaussian diameter distribution will have a lognormal molecular weight distribution.
Our method to estimate the molecular of NaYF4:RE UCNP populations will be of utility in a variety of applications, including the study of UCNP uptake by cells; separation of UCNPs by molecular weight, optical tweezing of UCNPs, and UCNP surface functionalisation for bio-imaging, bio-sensing, and therapeutic applications. Additionally we provide a stand-alone GUI to allow others to calculate 330 molecular weight of spherical NaYF4:RE UCNPs of arbitrary crystal lattice parameters, arbitrary diameters, and arbitrary RE dopant compositions.

Supplementary material: MATLAB Executable GUIs
Two stand-alone GUIs were created to enable calculation of UCNP molecular weight (the simple GUI, Figure S1), and the molecular weight of nanoparticles with arbitrary parameters (the advanced GUI, Figure S2). These GUIs are shown Figures S1 and S2. [Download Link]. NB: the final peerreviewed manuscript will link to a data repository with a DOI.

480
These GUIs were developed as stand-alone applications for Windows PCs (Windows 7 or newer) and are not compatible with Macintosh or Linux PCs. The GUIs were compiled into stand-alone 64bit executable files by using the MATLAB complier toolbox deploytool functionality. The GUIs were tested on 64-bit PCs running Windows 10. The first installation of a GUI was found to take around 45 minutes: this is primarily due to the automated download and installation of the required MATLAB runtime. After MATLAB runtime installation, the GUIs start-up time is typically < 10 seconds. Note that a MATLAB runtime compatible with MATLAB 2016a or newer is required to run these GUIs, and that MATLAB runtime versions older than 2016a may need to be removed from PCs prior to GUI installation.
490 Figure S1. The basic GUI developed to estimate the molecular weight of NaYF4:RE UCNPs of spherical geometry and cubic or hexagonal lattice parameters.

Competing financial interests
The authors declare no competing financial interests