Problems with CdSe(ZnS) quantum dots

Despite their potential for use as contrast agents for medical imaging techniques, their use in vivo is hindered by the cytotoxicity of cadmium. To address this issue, methods have been developed to “wrap” or “encapsulate” potentially-toxic QDs in bio-inert polymers to facilitate use in living tissue. While Cd-free QDs are commercially available, they are unsuitable for use as a substitute for organic contrasts.^[4] Another issue with CdSe(ZnS) nanoparticles is significant hydrophobicity, which hinders their ability to enter solution with aqueous media, such as blood or spinal fluid. Certain hydrophilic polymers could be used to render the dots water-soluble.

R_f-PEG synthesis

One notable quantum dot encapsulation technique involves utilizing a double fluoroalkyl-ended polyethylene glycol molecule (R_f-PEG) as a surfactant, which will spontaneously form micellular structures at its critical micelle concentration (CMC). The critical micelle concentration of the R_f-PEG depends on the length of the PEG portion of the polymer. This molecule consists of a hydrophilic PEG backbone with two hydrophilic terminal groups (C_nF_2n+1-CH₂CH₂O) attached via isophorone diurethane.^[5] It is synthesized by dehydrating a solution of 1,3-dimethyl-5-fluorouracil and PEG, mixing them in the presence of heavy water (D₂O) via a sonicator to combine then.^[6]

Diagram of R_f-PEG showing the fluoroalkyl end groups attached to the PEG polymer by intermediate groups.

Micellization

Polymer micelle with multiple QDs in the middle. The polymer wraps so both the hydrophobic ends are near the QDs and the hydrophilic end is on the outside of the micelle to allow solubility in water.

At the appropriate Krafft temperature and critical micelle concentration these molecules will form individual tear-drop loops, where the hydrophobic ends are attracted to one another, to other molecules, and also to the similarly hydrophobic QDs. This forms a loaded micelle with a hydrophilic outer shell and a hydrophobic core.^[6]

When encapsulating hydrophobes in this way it is important to ensure the particle size is appropriate for the PEG backbone being utilized, as the number of PEG mer units (generally with a molecular weight of 6 kDa or 10 kDa) determines the maximum particle size that can be successfully contained at the core of the micelle.

To determine the average diameter, D, of the QDs, the following empirical equation is used:

${\displaystyle {D=\ }{(1.6122\times 10^{-9})\lambda ^{4}}-{(2.66575\times 10^{-6})\lambda ^{3}}+{(1.6242\times 10^{-3})\lambda ^{2}}-{(0.4277)\lambda }+{41.57}}$

Where

• ${\displaystyle D}$ is the diameter of the CdSe QD in nm
• ${\displaystyle \lambda }$ is the wavelength of the first absorption peak in nm

Role of ZnS shell

It is during encapsulation that the ZnS shell plays an especially important role, in that it helps prevent the agglomeration of CdSe particles that had no shell by occupying the previously mentioned bonds on the dot's surface; however, clumping can still occur through secondary forces that arise from common hydrophobicity. This can result in multiple particles within each micelle, which may negatively impact overall resolution. For this reason multiple combinations of PEG chain length and particle diameter are necessary to achieve optimal imaging properties.

Micelles can be cross-linked. The ends of the polymer chain are attracted to two different QD groups.

Stokes-Einstein equation

The diffusivity of a particle decreases as its radius increases.

The diffusivity of spherical particles in a suspension is approximated by the Stokes–Einstein equation:^[6]

${\displaystyle D={k_{\text{B}}T \over \ 6\pi r\eta },}$ where

• ${\displaystyle T}$ is the temperature
• ${\displaystyle r}$ is the particle radius
• ${\displaystyle k_{\text{B}}}$ is the Boltzmann constant
• ${\displaystyle \eta }$ is the hydrogel viscosity

Typical R_f-PEG hydrogel diffusivities for 2 nm quantum dots are on the order of 10⁻¹⁶ m²/s, so suspensions of quantum dots tend to be very stable. Hydrogel viscosity can be determined by using rheological techniques.

Maxwell fluid

At molecular weights between 6 and 10 kilodaltons the R_f-PEG hydrogel acts as a Maxwell material, which means the fluid has both viscosity and elasticity. This is determined by measuring the plateau modulus, the elastic modulus for a viscoelastic polymer is constant or "relaxed" when deformed, at a range of frequencies via oscillatory rheology.^[9][10] Plotting the first- vs second-order integrals of the modulus values, a Cole-Cole plot is obtained, which, when fitted to a Maxwell model, provides the following relationship:

${\displaystyle {G^{\prime \prime }(\omega )=\ }[G^{\prime }(\omega )G_{0}-{G^{\prime }(\omega )}^{2}]^{-{1 \over 2}}}$

Where

• ${\displaystyle G_{0}}$ is the plateau modulus
• ${\displaystyle \omega }$ is the oscillation frequency in radians per second

Mechanical properties of common R_f-PEG molecules

Based on the Maxwellian behavior of the hydrogel and observations of erosion via surface plasmon resonance (SPR), the following data results for 3 common R_f-PEG types at their specified concentrations:^[11][12]

Property	10KC10	10KC8	6KC8
${\displaystyle {\text{Composition}}\ (wt\%)}$	6.8	6.5	11.0
${\displaystyle {\text{Relaxation Time}}\ (\tau _{r})\ (s)}$	1.2	0.029	0.023
${\displaystyle {\text{Plateau Modulus}}\ (G_{o})\ (kPa)}$	14.4	18.5	56.1
${\displaystyle {\text{Viscosity}}\ (\eta _{o})\ (kPa\cdot s)}$	18	0.53	1.5
${\displaystyle {\text{Conversion Percent}}\ (\%)}$	94	94	89

XKCY denotes X thousand daltons of molecular mass and Y carbon atoms.

These values can give us information on the degree of entanglement (or degree of cross linking, depending on what polymer is being considered). In general, higher degrees of entanglement leads to higher time required for the polymer to return to the undeformed state or relaxation times.