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Effervescent Redispersion of Lyophilized Polymeric Nanoparticles and the Physics of PEG
Steric-Layer Hydration on Aggregation
Carlos E. Figueroa1, Douglas H. Adamson2, and Robert K. Prud’homme1*
1Department of Chemical & Biological Engineering, Princeton University, Princeton, New
Jersey 08544
2Department of Chemistry and Institute for Material Science, University of Connecticut, Storrs,
Connecticut 06269
*Author for Correspondence, Tel: (609) 258-4577 Fax: (609) 258-0211 Email:
ABSTRACT
Background: Freeze-drying is an attractive method for converting nanoparticulate
pharmaceutical dispersions into a stable form with long shelf life. However, practical challenges
in translating laboratory practice to the medical setting, such as high cryoprotectant osmolarity
and infeasible reconstitution methods, currently limit lyophilized formulation development of
nanoparticle therapeutics.
Results: We demonstrate the use of effervescent redispersion for the reconstitution of lyophilized
model polymeric nanoparticles (O ~ 100 nm), which confers superior redispersability as
compared to the use of sucrose as a cryoprotectant. The effect of nanoparticle formulation
parameters (dispersion concentration, molecular weight of the stabilizing polymer, and physical
state of the nanoparticle core) on particle redispersability are examined and it is shown that 3:1
mass ratio of effervescent salt produces the optimum redispersibility. With only low-energy hand
agitation, redispersion to sizes less than 400 nm is achieved. The physics of compression and
2
hydration of the polyethylene glycol (PEG) steric layer protecting the nanoparticle surface is
derived. The calculations show the minimum value of the work required to compress the PEG
layer that is required for good redispersion and how this varies with PEG molecular weight and
surface density. .
Conclusions: This novel freeze-drying and reconstitution method offers an alternative solution to
the problematic traditional approaches to freeze-drying nanoparticles. In addition, general
guiding principles for the formulation of lyophilized polymeric nanoparticles have been
described.
KEY TERMS
Nanoparticles: Colloidal structures with characteristics sizes in the range of 1 to 1000 nm.
Freeze-Drying: A process in which a liquid solution or suspension is rapidly frozen and
subjected to a series of drying steps under vacuum to result in a powder.
Irreversible Aggregation: The fusion or coalescence of primary particles into a larger
agglomerate that requires significant energy input to break apart into the original particles.
Redispersability: The quality of being able to rehydrate a powder and yield suspended primary
particles without precipitation or settling.
Effervescence: The generation of carbon dioxide gas from an aqueous solution or suspension,
usually containing a carbonate or bicarbonate species.
Surface Coverage: A measure of how well covered a surface, such as that of a nanoparticle, is by
a stabilizing agent, such that two surfaces cannot contact each other directly.
ABBREVIATIONS
3
NP - nanoparticle
PEG - poly(ethylene glycol)
FNP - Flash NanoPrecipitation
PS - polystyrene
4
INTRODUCTION
There has been increasing interest in nanoparticulate constructs for drug delivery because
they can provide increased bioavailability, protection for the drug from degradation, prolonged
circulation, targeted delivery and/or extended release [1-7]. For many of these nanoparticles
(NPs), polymers are used to stabilize and modify the particle surface. Poly(ethylene glycol)
(PEG) is often used because of the excellent stabilization it provides, as well its ability to confer
resistance to opsonization in vivo [8].
It is generally desirable to dry the nanoparticle dispersions, as this can significantly
extend the shelf life of the formulations and the dry powders can be reconstituted when needed.
Since the therapeutic efficacy of a pharmaceutical nanoparticle formulation is associated with its
nanometer size, it is important to use techniques that avoid the irreversible aggregation of the
particles into micron-sized aggregates during drying. Usually, excipients are added to the
formulation to reduce aggregation. Nonetheless, there are constraints on the excipients that can
be used, since they should be biocompatible and they should not impart an excessive osmolarity
in the case of parenteral administration. Furthermore, excessive energy, such as sonication, that
cannot be replicated in a clinical setting should not be required to redisperse the powder back to
the primary particle size upon rehydrated. From these requirements it becomes clear that the
drying and reconstitution of nanoparticles is of great practical importance.
There are various techniques for drying nanoparticles; however, this work focuses on
freeze-drying. Briefly, this process involves rapidly freezing the nanoparticulate dispersion,
followed by lyophilization to sublime and remove the ice [9]. While the technique is generally a
lengthy batch operation, the benefits of the process include sterility, convenience, and low
5
temperatures (for drug stability). It is widely used in the pharmaceutical industry for formulating
drugs, peptides, and proteins [10].
However, reconstitution of lyophilized powders back to primary nanoparticles is a
challenge. During the freezing step, two main events contribute to particle aggregation:
mechanical stresses that occur during freezing can irreversibly force particles together, and the
growth of ice crystals can lead to concentration of the particles in unfrozen regions which may
induce aggregation [11]. To circumvent irreversible aggregation during freeze-drying,
cryoprotectants are introduced into the suspension that is to be frozen. The most commonly used
cryoprotectants are saccharides, such as glucose, sucrose, trehalose, etc. [9]. To achieve
acceptable redispersability, a high concentration of the sugar is required to dilute and entrap the
particles in a glassy matrix, thereby reducing aggregation. However, the high amount of
cryoprotectant creates problems in parenteral administration, as only a dilute concentration of the
active therapeutic can be administered before the osmolarity of the sugar makes the formulation
hypertonic. While there have been some attempts in the literature to use different types of
cryoprotectants, such as polymers that can be added at large mass ratios without making
suspensions hypertonic [11-15], there has not been work describing a cryoprotectant system that
imparts excellent redispersability to primary nanoparticles at acceptable osmolarities without the
need for energy-intensive methods, such as ultrasonication.
One method that could serve as a solution to this problem is the use of effervescent salts
as cryoprotectants. The motivation for using effervescence is that generation of gas bubbles can
impart energy at the primary particle level, bursting apart aggregates, and thus minimizing
external energy required to achieve acceptable redispersion. Effervescence is commonly used in
oral dosage formulations to achieve fast disintegration [16] as well as improved absorption of
6
drugs [17]. An effervescent spray-dried nanoparticulate formulation for inhalable respiratory
therapeutics has been designed by Ely et al. [18]. They demonstrated that when using an
effervescent carrier particle, embedded polybutylcyanoacrylate nanoparticles are able to be
redispersed to their original sizes. However, all redispersion work involved the use of sonication
for at least 1 minute and it is not clear what concentrations or ratios of effervescent salt to
nanoparticles were necessary.
In this study, we use nanoparticles produced using Flash NanoPrecipitation (FNP). This
process makes use of competitive time scales to precipitate both hydrophobic solute(s) or drug(s)
and an amphiphilic block copolymer from an organic stream into an aqueous anti-solvent [19].
Through rapid micro-mixing, the kinetics can be controlled such that the solute nucleates and
grows while the block copolymers self-assemble around the growing solute core. Aggregation is
arrested to produce nanoparticles with tunable sizes while stabilizing them with the hydrophilic
polymer block in the aqueous media. A dense steric protective layer on the nanoparticle surface
from the hydrophilic block of the polymer makes the particles biocompatible and long-
circulating [20]. Most of the nanoparticles described in this work were loaded with β-carotene,
which has previously been used as a model system for drug nanoparticles because of the
carotenoid's drug-like properties [19, 21, 22]. The nanoparticles are coated with polystyrene-
block-poly(ethylene glycol) (PS-b-PEG) block copolymers. Model compounds were used to
demonstrate the concepts controlling redispersion by this new technique. The key formulation
parameters are concentration of the nanoparticles and cryoprotectant, molecular weight of the
stabilizing polymer, and mechanical properties of the core.
EXPERIMENTAL
Experimental Reagents
7
D-α-tocopherol (97%), poly(propylene glycol) (average Mn ~3,500), and citric acid
(99%) were purchased from Sigma-Aldrich (St. Louis, MO) and used as received. Polystyrene-
block-poly(ethylene glycol) of 1.6K-b-1.8K and 1.6K-b-2.5K molecular weights were bought
from Polymer Source (Dorval, Quebec). Acetic acid (glacial, >99%) and sodium chloride
(>99%) were purchased from EMD Chemicals (Darmstadt, Germany). Tetrahydrofuran (99.9%)
and sodium bicarbonate (>99%) were purchased from Fisher Scientific. β-carotene (99.9%) was
received from BASF (Ludwigshafen, Germany). Polystyrene (PS) and PS-b-PEG (1.5K-b-5.3K)
were synthesized by high vacuum anionic polymerization methods [23] using hydroxide capped
PS initiated by potassium naphthalenide followed by addition of ethylene oxide. Deionized water
treated by passage through a series of ion exchange columns to a final resistance of ≥17.9 mΩ
will be referred to as MilliQ water.
Nanoparticle Formation
The nanoparticles were produced through the FNP process, using a laboratory-scale
confined impinging jet mixer [24]. The active core material(s) and the block copolymer were
dissolved in tetrahydrofuran to form the organic phase. The organic phase was manually injected
into the mixer against an equal volume of MilliQ water using 1mL plastic syringes (National
Scientific Company, Rockwood, TN) with a syringe jet velocity of ~35 m/s and the effluent was
collected into a water reservoir, resulting in a tetrahydrofuran:water ratio of 1:9. The particle size
of the dispersion was measured and different aliquots were freeze-dried with or without
cryoprotectants.
Nanoparticle Characterization
Particle size distributions were measured via dynamic light scattering using a ZetaSizer
Nano ZS ZEN 3600 (Malvern Instruments, Worcestershire, U.K.). Each measurement was
8
repeated on a diluted sample at least twice and the particle size reported is the diameter
corresponding to the peak of the intensity-weighted average size distribution, as reported by the
Zetasizer software through a normal resolution analysis mode.
Freeze-Drying of Nanoparticles
Freeze-drying consists of rapid freezing and subsequent lyophilization of the dispersion.
To achieve rapid freezing, an aliquot of ≤3 mL of the dispersion was placed in a 5 mL cryogenic
vial (VWR International, Radnor, PA) which was submersed in an acetone/dry ice bath at -78 °C
for at least 30 minutes. Lyophilization was performed on a Benchtop 3.3/Vacu-Freeze (VirTis,
Gardiner, NY) under vacuum (< 100 mTorr), with a condenser temperature of -78 °C. After at
least 1 day, the dried powders were removed from the lyophilizer and analyzed within an hour
after removal.
Reconstitution of Lyophilized Nanoparticles
In order to resuspend freeze-dried nanoparticles, a dilute aqueous acid (15-25 mM) was
added to the cryo vials to achieve the concentration desired. The final pH of the dispersion was
close to neutral (6.5 ≤ pH ≤ 7.5). The vials were then either agitated manually (~180-210 shakes
per minute) or sonicated using a probe-tip ultrasonicator for one minute. The particle sizes were
then measured to determine the redispersability.
Imaging of Nanoparticles
Scanning electron microscopy was used to obtain micrographs of the nanoparticles. An
XL30 FEG-SEM digital scanning microscope (FEI, Hillsboro, OR) was used with a cold field
emission cathode operated at 5 kV. Prior to imaging, the NPs were freeze-dried and applied to
carbon tape. The samples were then coated with a 5 nm thick iridium layer.
Statistical Analysis
9
All sample t-test analysis was done using the built-in statistical tools in Origin 8 software
(OriginLab Corporation, Northampton, MA).
pH Equilibrium Analysis
Analyses of bicarbonate pH equilibria were done using CurTiPot pH and Acid-Base
Titration 3.5.4 for Excel [25].
RESULTS AND DISCUSSION
Proof of Concept
To determine the effectiveness of sodium bicarbonate as a cryoprotectant and the
subsequent effervescence upon reconstitution, β-carotene nanoparticles were freeze-dried with
various concentrations of NaHCO3 (Fig. 1).. A mass ratio of 3:1 sodium bicarbonate to
nanoparticles was used for subsequent comparisons since the data show that at ratios higher than
3:1, the redispersability ratio (redispersed size over initial size) was not further reduced (within
statistical significance of p > 0.05). The lowest amount of salt was used in order to minimize the
osmolarity of the formulation upon reconstitution and to avoid precipitation of the PEG steric
layer at high ionic strengths [26]. Triplicate samples were reconstituted to evaluate
reproducibility of the results.
Figures 2a and 2b show the redispersed particle size distributions of the nanoparticles
with and without NaHCO3, respectively. The data show the redispersion after sonication and
after hand agitation. No data are shown for the hand agitation of the powders without NaHCO3
because macroscopic precipitates were visible and dynamic light scattering measurements were
not possible. However, sonication was able to reconstitute these powders to particles in the nano
range (Sf/Si = 2.4±0.1). The effervescent powders showed improved redispersability; hand
10
agitation was able to redisperse particles to sizes similar to the sonicated bare powders (Sf/Si =
2.9±0.3). When sonication was employed, these particles redispersed essentially to the initial size
(Sf/Si = 1.2±0.1).
SEM imaging of the lyophilized powders revealed that at low magnification, there was a
difference in morphology between the particles (Figures 3a, 3c). The powder without salt had
macroscopic aggregates and it was difficult to see any small particles that could be identified as
primary nanoparticle clusters. Aggregates (Figure 3d) were on the order of magnitude of a
micron. Contrastingly, the effervescent powder had much finer detail, where no large primary
particle aggregates. High magnification revealed sub-micron particles embedded in the salt
matrix (Figure 3f), which correlates with the superior redispersability of the effervescent powder.
While it is clear that the effervescent salt helped to prevent aggregation, it is not certain
from these results whether it is only the matrix formation mechanism that was the cause or if the
actual effervescence helped to burst apart particles during reconstitution. To address the
mechanism of action, freeze-drying was done comparing nanoparticle dispersions with sodium
chloride, which is not effervescent, and sodium bicarbonate, both at the 3:1 mass ratio. This
allowed the differences in results to be attributed to effervescence. When reconstituted, Figure 4a
shows that the NaCl powder redispersed to approximately 380 nm, which is larger than the
effervescent powder size of 200 nm. Therefore, it can be inferred that while the salt acted as a
cryoprotectant in reducing particle aggregation, the effervescence aided in breaking apart larger
aggregates. This is consistent with the SEM micrographs comparing the two salt powders in
Figures 3b and 3c. The two salt powders were similar at both low and high magnifications
(Figures 3e and 3f).
11
From these preliminary results, it is evident that sodium bicarbonate can reduce
nanoparticle aggregation during freeze-drying. D'Addio and co-workers previously found that to
properly redisperse β-carotene nanoparticles stabilized by poly(ethylene glycol)-block-
poly(lactide-co-glycolide), a 60:1 mass ratio of sucrose to nanoparticles was necessary [27].
When comparing the use of the effervescent cryoprotectant to that of sucrose , β-carotene
nanoparticles redispersed to a final concentration of 2 mg/mL achieved a lower osmolarity at a
shorter sonication time with sodium bicarbonate (~160 mOsM; 1 minute) relative to sucrose (350
mOsM; 15 minutes sonication). Furthermore, even hand agitation of the effervescent formulation
yielded an acceptable redispersability ratio less than 4. Since the goal was to produce sub-600
nm particles using hand agitation for reconstitution in medical settings, all subsequent trials
avoided the use of sonication.
Effervescent Formulation Optimization
To optimize the effervescent reaction conditions, the acid used and the final pH of the
dispersion were varied. The two acids that were evaluated, acetic and citric, gave equivalent
redispersion as shown in Figure 4b. There was no statistically significant difference in particle
size when comparing the two acids (p > 0.05). Citric acid was used in all subsequent trials as a
matter of convenience. Additionally, since the effervescent reaction of bicarbonate generating
carbon dioxide gas is pH dependent (see reaction below), the effect of pH was investigated.
( ) ( )( )
( ) ( ) ( )
( ) ( )
2H O3 3
23 2 3 3
2 3 2 2
2 2
R COOH HCO R COO H HCO
H HCO H CO 2H CO
H CO CO H O
CO CO
aq X s X
aq
aq aq l
aq g
− + + −
+ − + −
− + → − + + +
→ →+ +← ←
→ +←
→←
12
Based on the equilibrium of the carbonate, bicarbonate, and carbon dioxide species, the dominant
species depends on the pH. Figure 4c shows that acidic pH is necessary for generation of
significant gas. However, since the motivation for this effervescent formulation is parenteral
administration of the nanoparticle dispersion at neutral pH, two different neutralization schemes
were tested: titration of the powder to near neutral pH, and titration to pH~4 (acidification)
followed by neutralization. In both cases, the final concentration of nanoparticles was the held
constant. Figure 4d shows that the redispersed particle sizes were not statistically different (p >
0.05). At pH~4, where essentially all of the bicarbonate has evolved as gas, the particle size is
equivalent to the neutral pH case, where only about 20% of the bicarbonate has evolved into gas
form. Since no enhancement in redispersability is afforded by lower pH, immediate neutral pH
titration was used for all further trials.
Overall, while it seems that the effervescent generation of gas bubbles produces enhanced
redispersability, the effect is not sensitive to the details of neutralization. This finding allows for
the technique to be amenable to applications where different final pHs and ionic strengths, as
well as specific ions, are needed. However, if the technique were to be advanced to the clinic,
additional work on toxicology, drug stability, and pharmaco-kinetics would be required.
NP Formulation Optimization
The effects of several nanoparticle formulation parameters on the particle redispersability
were considered. Specifically, the concentration of the nanoparticle dispersion, molecular weight
of the stabilizing hydrophilic block of the block copolymer, and core hardness were studied.
Concentration of NP Dispersion
13
The concentration of the dispersion will affect how concentrated the particles become
during cryo-concentration in the freezing step. β-carotene nanoparticles were made at a base
concentration of 2 mg/mL and an aliquot was diluted to 1 mg/mL while another was
concentrated to 3.56 mg/mL by using external Drierite calcium sulfate dessicant (W.A.
Hammond Drierite Company, Xenia, OH) to remove water from a sample contained in dialysis
tubing (MWCO: 6-8,000) (Spectrum Laboratories, Rancho Dominguez, CA). Samples at the
three concentrations were freeze-dried using a 3:1 salt to nanoparticle mass ratio. Reconstitution
was done via hand agitation for 1 minute. Figure 5 shows that the more concentrated the
nanoparticle dispersions, the larger the size of the redispersed particles. The particle size
increased from 58 nm to 310 nm for a 1 mg/mL dispersion and from 67 nm to 700 nm for a 3.56
mg/mL dispersion, both freeze-dried with a 3:1 NaHCO3:NP mass ratio.
At higher initial concentrations, it is more probable that thermal stresses will force
primary particles together upon freezing. Also, the unfrozen regions that undergo cryo-
concentration will experience an increased probability of particles contacts. This higher
probability of particles contacting increases aggregate formation at higher nanoparticle
concentrations.
Molecular Weight of Stabilizing PEG Polymer
For investigating the effects of molecular weight of the block copolymer, a series of β-
carotene nanoparticles were produced. Three different PS-b-PEG polymers were used, each with
a similar polystyrene hydrophobic block but with poly(ethylene glycol) blocks of increasing
lengths: 1.6K-b-1.8K, 1.6K-b-2.5K, 1.5K-b-5.3K. Using mass ratios of block copolymer to β-
carotene of 12/8, 10/10, 7/13, 5/15, 3/17 in the organic stream at total solids concentrations of 20
mg/mL, different size nanoparticles were made at the same final concentration of 2 mg/mL. The
14
particle diameters varied from 62 nm to 197 nm as the block copolymer fraction varied from
60% to 15%, as is shown in Figure 6. This dependence of particle size on block copolymer
concentration arises because in FNP the growth of β-carotene nuclei is kinetically stopped by the
assembly of the amphiphilic polymer on the nuclei surface. Surface coverage arrests further
aggregation. Thus, an abundance of the block copolymer, relative to the core solute, results in
less growth [28].
For each size nanoparticle, duplicate samples were prepared with 6 mg/mL sodium
bicarbonate (3:1 salt:NP mass ratio) and 3 mL aliquots were freeze-dried. The dried powders
were then effervescently redispersed and particle size measurements were taken. The results are
presented in Table 1, in terms of initial and redispersed particle diameters and redispersability
ratios. The redispersion of the various formulations by hand agitation varied significantly,
producing redispersability ratios from 2.2 to11.9 depending on the particular particle
composition.
Physics of Polymer Brush Repulsions on Redispersion Efficiency
To understand the results of the reconstitution, consider the freeze-drying process.
Freezing and lyophilization involve the following physical processes. Since the bicarbonate is
not a glass former, the freezing process involves water crystals concentrating the salt and
nanoparticles until salt precipitation begins. The salt crystals act as mechanical barriers keeping
particles from contacting each other, and can also force primary particles together during drying
when there are no salt or water crystals between them. Hence the dependence of redispersion on
both the concentration of the particles and the concentration of bicarbonate. Nanoparticles that
are kept apart remain so during the lyophilization process, but particles that are forced together
are brought into even more intimate contact when the ~85% water in their brush layers is
15
removed. The final state involves some fraction of the nanoparticles in intimate contact. During
reconstitution, water rehydrates the PEG brush layer and an osmotic force between the two brush
layers tends to separate and redisperse the chains. It is this process and the details of the brush
layer that lead to the differences in redispersion that are observed experimentally.
The process of redispersion involves hydration of the compressed PEG layer formed
during the drying process. The following model of the polymer physics captures the results
observed experimentally. The details of the theory and its development are given in the
Supplementary Information. First, the thickness of the PEG corona and the number of chains per
surface area are calculated for each particle system, since the chain density and molecular weight
determines osmotic repulsion during rehydration. Refer to Table S2 in the Supplementary
Information for the specific properties calculated and their values for each particle batch. The
PEG chains on the nanoparticle surface adopt one of two conformations. If the chains are
separated on the surface and non-interacting, then their size is that of an ideal polymer in
solution. This is called the “mushroom” regime, which has a more diffuse polymer layer and
greater exposure of the core [29]. If the chains are closely packed on the surface, their repulsions
cause the layer to expand such that its thickness becomes larger than the dimension of a single
chain in solution. This is called the “brush” regime, which has a dense polymer layer that
provides better screening of the core [29]. In comparing the results for each nanoparticle
formulation, the experimentally determined chains per surface area, σ, are closer to the
equilibrium brush regime than the mushroom regime. Since the FNP assembly is a kinetically
arrested process, rather than an equilibrium process, the experimental PEG density is somewhat
less than what would be achieved if equilibrium were achieved. This has been shown in
simulations of kinetically determined micelle assembly [30].
16
From these calculated values, the osmotic pressure that the PEG chains exert as function
of separation distance is calculated and the work required to compress the steric layers is
determined. To calculate the repulsive work from the compression of the PEG layers against
their osmotic pressure, the analysis of PEG brush layers of Hansen et al. was used [31]. The
osmotic pressure as a function of separation distance between two hydrophobic cores, x, is given
below where α is a constant determined by Hansen to be 0.8 for PEG, kB is the Boltzmann
constant, T is temperature (where T = 298 K is assumed), b is the monomer length, N is the
number of monomers in the chain, and δexp is the experimentally calculated polymer layer
thickness.
( )3/49/2 9/4
exp exp3
exp
Bk T xxb bN x
δ δα
δ
Π = − (1)
The work Wsteric is calculated by integrating the osmotic pressure through the compression of the
layers from the point of first contact (δexp) to the point where the two cores are separated by a
distance equal to the thickness of a dehydrated PEG layer, ℓ. This is shown schematically in
Figure 7. The thickness of the dehydrated polymer layer is estimated by assuming that the
volume of dry PEG on a particle will uniformly coat a spherical core and thus ℓ can be
determined by subtracting the core radius (from the PEG layer thickness calculation) from the
condensed particle radius. The chain density on the surface is taken as constant and the
Derjaguin approximation is used to calculate the force between the two approaching spheres to
obtain Equation 13, where A is the contact area between two particles and DNP is the particle
diameter.
( ) ( ) ( )exp exp exp
22
exp4 2NP NP
stericD DW V x dV x Adx x x dx
δ δ δπ δ
= Π∆ = Π = Π = Π − − +
∫ ∫ ∫
(2)
17
Figure 8 shows the correlation between this calculated work, or the work necessary for
complete compression of the polymer layer, and the experimentally observed redispersability
ratio. From the figure, or the calculated work values in Table 1, it is clear that the higher
molecular weight PEG chains required larger osmotic work values for compression. This
correlates with better redispersability. The data for all of the formulations with different PEG
densities and particle sizes collapse well. The result shows that repulsive work greater than 10-18
J per particle contact was necessary to produce redispersability ratios lower than 4. This value of
4 or lower is a target for redispersion since particles less than 400 nm are generally desirable for
IV injection and 100 nm particles are generally achievable by FNP. Since all particles formed
from 5.3K PEG block copolymer were able to achieve these redispersability ratios, it is
recommended that this PEG molecular weight be used with this 3:1 loading of sodium
bicarbonate as cryoprotectant. It is expected that if no cryoprotectant were used, the same trend
would exist, but the redispersability ratios would be much higher. Furthermore, it is also shown
that larger particle sizes yielded higher calculated osmotic work (Table 1), as well as lower
experimentally observed redispersability ratios. In comparing formulations using the same block
copolymer, Equations 1 and 2 indicate that while all formulations should result in similar
osmotic pressures between polymer layers, the contact area will be larger for the larger particles,
thus increasing the repulsive work. This appears to be the mechanism whereby better
redispersability is achieved for larger particles.
As a secondary calculation, we calculated the attractive van der Waals interactions
between two nanoparticle cores to estimate the relative importance of purely attractive
interactions to the repulsive osmotic interactions of the PEG. For the complete methodology
used, please refer to the Supplementary Information. The attractive work Wvdw was calculated for
18
each particle batch from an initial point where two particles have the PEG coronas touching
(2δexp) to the point where the PEG layers have been condensed and the cores are separated only
by a dehydrated PEG layer (2ℓ) (Figure 7). An example comparing the attractive and repulsive
work is given in Figure 9.
The results indicate that from separation distances of ℓ, i.e. the thickness of the uniformly
condensed PEG layer, to δexp, the osmotic force is always greater than the attractive van der
Waals interactions. However, it is experimentally observed that with minimal energy during
redispersion, there is always some aggregation. With more intense sonication, the aggregates can
be dispersed. What is the reason that aggregation is observed? There are two phenomena not
considered in this model that are probably the source of the discrepancy. First, the PEG layer is
assumed to be uniform so that there is no direct hydrophobic interaction between the two
nanoparticle cores. The model calculation assumes the particle contacts occur on the PEG layer
that can be hydrated. In actuality, there are likely to be fluctuations in PEG density on the
surface, exposing the hydrophobic cores at times so that some core-core contacts can occur and
lead to aggregation. Without the hydratable PEG layer between surfaces, the osmotic repulsion
of the PEG may be too weak to enable aggregate break-up after lyophilization. Also, the growing
ice and salt crystals may exert sufficient mechanical force on contacting particles to cause
displacement of the PEG layer and thus force direct core-core interactions. Calculating the
probabilities of these occurrences or the magnitudes of the effects is beyond the scope of this
study. The goal of the modeling was to capture the main features of the role of polymer
molecular weight during freeze-drying of nanoparticles, which we believe it does.
Material Properties of Core Solute
19
In addition to the role of the PEG block molecular weight, the properties of the
nanoparticle core also influence redispersability. Rigidity or softness of the nanoparticle core can
make a difference in terms of the dynamics of the stabilizing chain on the particle surface, as
well as on the stress distribution as primary particles are forced together by water crystallization.
A series of nanoparticles were made using the 1.5K-b-5.3K PS-b-PEG and with either a solid
(i.e. high Tm or Tg) or a liquid (i.e. low Tg) core. They were also chosen so that in each class
there is either a low molecular weight solute or a polymeric core solute. Each particle
formulation was made at a total solids concentration of 20 mg/mL in the organic stream, using
mass ratios of block copolymer to core solute of 12/8, 7/13, and 3/17. The cores evaluated were
β-carotene (crystalline solid), D-α-tocopherol (liquid), polystyrene 1.5K (amorphous solid at
25°C), and poly(propylene glycol) 3.5K (amorphous liquid at 25°C). The nanoparticles were
freeze-dried using a 3:1 mass ratio of salt to particles at a particle concentration of 2 mg/mL.
Effervescent reconstitution was done using 1 minute of hand agitation.
The results are displayed in Figure 10, where the redispersability ratios are plotted versus
the PEG osmotic work. The redispersability ratios for the solid cores are consistently larger than
those of the liquid cores at the same calculated osmotic work. This outcome can be understood
by considering the way in which the nanoparticles will behave when subjected to the stresses
characteristic of the freezing step. The solid cores, β-carotene, which has been shown to be
crystalline when nanoprecipitated [32], and polystyrene, which is glassy [33], are not able to
deform during the compression of two or more primary nanoparticles. Thus, mechanical forces
are concentrated at a small area, which creates large local stresses. This may cause displacement
of the protecting PEG steric chains as was hypothesized above. This results in bridging and/or
coalescence of the primary particles [34]. In contrast, the liquid core particles are capable of
20
deforming, which allows for the force to be distributed over a much larger surface area. The
larger area reduces the stress and also prevents local rearrangements of the PEG chains. This
phenomena is observed in the coalescence of polymer stabilized drops in polymerization
processes. Initially, the low viscosity droplets resist coalescence because they are deformable; as
polymerization progresses, they become stiffer and less deformable, with the result that
collisions result in particle coalescence.
CONCLUSIONS
This work has demonstrated the use of effervescent redispersion for reconstitution of
lyophilized polymeric nanoparticles. Using sodium bicarbonate as the cryoprotectant and then
adding acid to evolve gas enhances redispersion back to the primary particle size. At ratios of
sodium bicarbonate to nanoparticle of 3:1, the original 100 nm β-carotene nanoparticles were
resuspended to ~300 nm in size with hand agitation. The comparison of a non-effervescing salt,
NaCl, to the effervescent sodium bicarbonate shows that the effervescence contributed to the
redispersion. The redispersion is not attributable to merely particle separation and dilution by the
cryoprotectant. However, freeze-drying at lower nanoparticle concentrations, i.e. dilution, does
decrease aggregation.
Hand agitation for 1 minute represents the energy input in a medical drug administration
application, and ultrasonication represents a high level of energy input for maximum
redispersion. Hand agitation for one minute of a reconstituted sample with NaHCO3 yields a
redispersability ratio (i.e. redispersed particle size relative to the initial particle size before
freeze-drying) similar to that of a sample freeze-dried sample without cryoprotectant subjected to
one minute of sonication. For the mass ratio of sodium bicarbonate to nanoparticles (3:1), one
21
minute of sonication after titration to neutral pH results in a particle size distribution very close
to the initial prior to freeze-drying. High energy input can completely redisperse these PEG-
protected nanoparticles.
Several factors associated with the block copolymer steric protective layer have been
identified as having a significant impact on the redispersability. The rapid precipitation process
that we have used to create these nanoparticles, Flash NanoPrecipitation, produces nanoparticles
with dense PEG layers in the brush regime. These PEG layers both enhance in vivo circulation of
the particles, and also contribute to the osmotic forces that separate nanoparticles during freeze-
drying. Calculations were performed to estimate the osmotic work necessary to compress the
polymer layers of two particles in direct contact during freeze-drying, or equivalently the work
available to separate two lyophilized particles that are in direct contact during reconstitution. The
calculations indicate that osmotic work greater than 10-18 J in magnitude produced
redispersability ratios below 4. The redispersability ratio correlated with the osmotic work for the
three block copolymers studied and over the range of particle sizes from 62 nm to 197 nm. PEG
chains of molecular weight of 5.3K yielded acceptable redispersability by hand agitation
regardless of the particle size. Lastly, the rigidity of the particles influences how the inter-particle
stresses produced during freezing induce aggregation. Nanoparticle formulations with fluid-like
cores redispersed to within 50% of their initial size when formulated using the 5.3K PEG block
copolymer. The use of liquid core components, such as α-tocopherol or poly(propylene glycol),
are attractive routes to low energy redispersability.
Effervescent redispersion is shown to be an effective technique for processing PEG-
protected nanoparticles. The rules for formulating redispersible lyophilized powders include
using mass ratios for effervescent agent to nanoparticles of 3:1, having dense PEG steric layers
22
afforded by 5K PEG block copolymers, optimizing particle concentration, and controlling the
core physical properties.
FUTURE PERSPECTIVE
Considering the increasing development of and the need for pharmaceutical
nanoparticulate therapies, it is crucial to institute practical freeze-drying and reconstitution
methods that are feasible in a medical setting. The formulations presented here avoid freeze-
drying with large amounts of cryoprotectants that would yield hyperosmotic dispersions in order
to achieve high drug concentrations for parenteral injection. While we have primarily focused on
a proof of concept, there would be additional stability and toxicology issues to address for further
development. The fact that the individual ingredients in this effervescent redispersion study are
approved for parenteral administration is a positive point in favor of further development.
Most of the focus today on drug nanotechnology is on the development of delivery
vehicles for therapeutics, but there has been relatively little work on nanoparticle processing.
Successful introduction of nano-therapeutics will require both nanoparticle design and
processing. Lyophilized nanoparticle powders will be the most likely form for commercial
therapeutics. It is with this motivation that this work has been done, in hopes that these findings
can serve as guide for future formulations that require freeze drying.
SYMBOLS USED
Sf/Si redispersability ratio or the ratio of the final redispersed size to the initial size ξ blob size of a polymer chain
kB Boltzmann constant T absolute temperature N number of monomers in a polymer chain
23
b monomer length σ surface coverage, or the number of polymer chains per surface area on a surface δ polymer layer thickness DNP diameter of a nanoparticle p aggregation number k a constant x separation distance l thickness of a dehydrated polymer layer W work
EXECUTIVE SUMMARY
Introduction
• Freeze-drying and reconstitution of nanoparticles is challenging because of the restrictions on
the use of cryoprotectants, which include biocompatibility and low osmolarity. Furthermore,
reconstitution should be such that requires the lowest energy input.
Proof of Concept
• Effervescent redispersion, or reconstitution of a lyophilized nanoparticulate powder
containing a bicarbonate salt, to neutral pH has been shown to yield better redispersability at
a lower osmolarity than reconstitution of powders containing sucrose.
Effervescent Formulation Optimization
• Parameters for effervescence, such as acid choice and pH of redispersion, do not have a
significant effect on the redispersability of nanoparticles.
NP Formulation Optimization
• The concentration of nanoparticles in the dispersion that is to be freeze-dried has a large
effect on the redispersability, where dilute samples yield smaller redispersed particle sizes.
24
• Using Flash NanoPrecipitation, the nanoparticles used in this study exhibit dense steric PEG
layers that are closer to being in an equilibrium brush conformation than a mushroom
conformation.
• For better redispersability, it is recommended to use a block copolymer with a larger
molecular weight PEG block to increase the osmotic work and thus the repulsion between
particles.
• The repulsive osmotic pressure work of the steric PEG layer of a nanoparticle is much
greater than the attractive van der Waals work between two particle cores.
• In comparing cores with solid materials (i.e. glassy or crystalline) to liquid cores,
nanoparticles with liquid cores consistently redispersed better, which is most likely due to the
way in which stress distributes through the particles.
25
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REFERENCE ANNOTATIONS
**9. This review on freeze-drying of nanoparticles has a great overview on what cryoprotectants
have been attempted in the literature.
**18. The use of effervescence in spray dried formulations of nanoparticles is investigated and
provides the motivation for this work.
*19. The details of the controlled assembly process in Flash NanoPrecipitation are described.
*27. A comparative study between different forms of drying nanoparticles is presented and
highlights some of the difficulties in freeze-drying.
29
**31. Based on bulk and lipid bilayer data, the osmotic pressure of PEG brush layers is evaluated
and explained.
ACKNOWLEDGMENTS
This work was made possible through support from the National Science Foundation
through the NIRT “Nanoscale Interdisciplinary Research Teams” (CTS-0506966).
30
CAPTIONS FOR FIGURES/TABLES
Figure 1 Optimization of mass ratio of sodium bicarbonate to β-carotene nanoparticles. Using
PS-b-PEG (1.5K-b-5.3K) stabilized β-carotene NPs at 2 mg/mL, the amount of sodium
bicarbonate added prior to freeze-drying was varied. Reconstitution of the dry powders back to
their original concentration was achieved by rehydration with dilute aqueous acetic acid to
produce neutral pH dispersions. All samples were then hand-agitated for 1 minute. For sodium
bicarbonate, there is no significant difference in average particle size between 3:1 and 10:1 mass
ratios (p > 0.05).
Figure 2 Reproducibility of reconstitution of lyophilized nanoparticles containing either (a) no
cryoprotectant or (b) a 3:1 mass ratio of sodium bicarbonate to NPs. In each plot, there are at
least two replicates presented. Symbols: () - initial NP dispersion, () - particles reconstituted
via hand agitation for 1 minute, () - particles reconstituted via probe tip ultrasonication for 1
minute. Note: In Figure 2a, no data is shown for reconstitution via hand agitation because the
sample had macroscopic aggregates making the measurement unrepresentative.
Figure 3 SEM micrographs of lyophilized particle powders. (a) - (c) are at 1000x magnification,
while (d) - (f) are at 20,000x. The cryoprotectant used is as follows: (a) and (d) none, (b) and (e)
sodium chloride, and (c) and (f) sodium bicarbonate. The added scale bar in (a) is valid for (a) -
(c) and the scale bar in (d) is valid for (d) - (f).
31
Figure 4 Effect of effervescent versus non-effervescent salts on NP redispersability and effect of
acid type and pH on NP redispersability. (a) Comparison of reconstitution of NPs freeze-dried
with sodium bicarbonate vs. sodium chloride at a 3:1 salt to particles mass ratio. Symbols: () -
initial NP dispersion, () - sodium bicarbonate, (∆) - sodium chloride. (b) Comparison of
reconstitution of NPs freeze-dried with sodium bicarbonate that were neutralized using acetic
acid or citric acid. There is no significant difference in average particle size for the two acids (p
> 0.05). Symbols: ()- initial NP redispersion, () - acetic acid as neutralizing agent, (∆) - citric
acid as neutralizing agent. (c) Chemical equilibrium of the bicarbonate species as a function of
pH. Lines: solid - carbonate ion, dashed - bicarbonate ion, dotted, - carbon dioxide gas. (d)
Particle diameters of redispersed lyophilized particles that were either neutralized from the start
(solid bars) or were reconstituted to pH~4 and subsequently neutralized (striped bars). There is
no significant difference in particle size for both neutralization schemes (p > 0.05). All
reconstitution was by 1 minute of hand agitation.
Figure 5 Effect of initial NP concentration on redispersed particle size. β-carotene NPs were
freeze-dried with sodium bicarbonate at a mass ratio of 3:1 salt to particles and reconstituted
back to their original concentrations by hand agitation for 1 minute. The solid black bars
represent the initial NP dispersion and the open striped bars indicate the effervescently
redispersed NPs.
Figure 6 Dependence of β-carotene NP diameter on block copolymer weight percent in the
organic stream used for FNP. The block copolymer used was PS-b-PEG. The symbols represent
32
the following polymer molecular weight: () - 1.6K-b-1.8K, () - 1.6K-b-2.5K, () - 1.5K-b-
5.3K.
Table 1 Results of β-carotene NP production and effervescent redispersion of lyophilized NP
powders. All samples were produced with a total solids concentration of 2 mg/mL and were
freeze-dried with a 3:1 sodium bicarbonate to NP mass ratio. Reconstitution was done via hand
agitation for 1 minute. Sample name codes are as follows. All block copolymers were PS-b-PEG,
with a 1.6K PS block (except for the 1.5K-5.3K) and thus only the molecular weight of the PEG
block is listed, followed by the mass ratio of block copolymer to β-carotene in the NP
formulation: 12/8, 10/10, 7/13, 5/15, 3/17. The initial diameters reported are the average of
triplicate measurements, while the reported redispersed diameters are the average of at least
duplicate measurements of three samples followed by the standard deviation. The
redispersability ratio is the ratio of redispersed diameter to initial diameter.
* No redispersed diameter or redispersability ratio is reported because the dynamic light
scattering measurements for these samples were not consistent.
** Only one measurement yielded reliable data and thus no standard deviation is reported.
Table 2 Characteristics of hydrated PEG chains of different molecular weights in the
experimental nanoparticle system, as well as in the mushroom and brush conformations. The
sample name code is the same as in Table 1.
Figure 7 A simple schematic of two NPs with the steric PEG layers being compressed. The
initial PEG brush layer thickness, δexp, is present when the two particles first touch. The brush
33
layers are subsequently compressed, assuming constant chain density on the core surface, until
the two cores are separated by a thickness equal to twice the thickness of dehydrated PEG layer,
ℓ.
Figure 8 Effervescent redispersion of β-carotene NPs stabilized with PS-b-PEG block
copolymers of similar PS block molecular weights and varying PEG block molecular weights.
The various symbols denote the different PS-b-PEG block copolymers used: () - 1.6K-b-1.8K,
() - 1.6K-b-2.5K, () - 1.5K-b-5.3K. All samples were hand-agitated for 1 minute. The plot
compares the redispersability ratio (final redispersed size to initial size Sf/Si) to the work required
to compress the PEG layer separating two touching particles from the point of first contact to a
separation distance equal to the dehydrated thickness of the PEG corona.
Figure 9 A comparison of the repulsive and attractive forces as particle cores become closer
through PEG layer compression of NP formulation PEG1.8K: 12/8. The range of x/δexp in the plot
corresponds to ℓ/δexp to 1, which is the range considered for the osmotic work calculation. In
calculating the forces, the repulsion is that associated with the osmotic pressure of the PEG
corona (solid line) and the attraction is taken as the van der Waals forces (dashed line). See text
for assumptions and equations used.
Figure 10 Effervescent redispersion of nanoparticles stabilized by PS-b-PEG (1.5K-b-5.3K)
block copolymer and various core solutes. The symbols for the core solutes are as follows: () -
β-carotene, () - 1.5K polystyrene, () - D-α-tocopherol, () - 3.5K poly(propylene glycol).
All samples were hand-agitated for 1 minute. The plot compares the redispersability ratio (final
34
redispersed size to initial size Sf/Si) to the work required to compress the PEG layer separating
two touching particles from the point of first contact to a separation distance equal to the
dehydrated thickness of the PEG corona.
Figure 1
Figure 2
Figure 3
Figure 4
Figure 5
Figure 6
Figure 7
Figure 8
Figure 9
Figure 10
SUPPLEMENTARY INFORMATION
Effervescent Redispersion of Lyophilized Polymeric Nanoparticles
Carlos E. Figueroa1, Douglas H. Adamson2, and Robert K. Prud’homme1*
1Department of Chemical & Biological Engineering, Princeton University, Princeton, New
Jersey 08544
2Department of Chemistry and Institute for Material Science, University of Connecticut, Storrs,
Connecticut 06269
*Author for Correspondence, Tel: (609) 258-4577 Fax: (609) 258-0211 Email:
ABBREVIATIONS
PEG - poly(ethylene glycol)
NP - nanoparticle
BCP - block copolymer
PS - polystyrene
SMD - surface area moment mean diameter
POLYMER BRUSH PHYSICS CALCULATIONS
The poly(ethylene glycol) (PEG) chains on the nanoparticle surface adopt one of two
conformations. If the chains are separated on the surface and non-interacting, then their size is
that of a random coil in solution. This is called the “mushroom” regime [1]. If the chains are
closely packed on the surface, their lateral excluded volume repulsions cause the layer to expand
such that its thickness becomes larger than the dimension of a single chain in solution. This is
called the “brush” regime [1]. In the brush regime, the density of the chains at the surface is
determined by the balance between the stretching energy of the chains in the brush and the
energy required to create the unfavorable hydrophobic interface between the aqueous phase and
the hydrophobic core of the NP. The calculations of PEG densities on nanoparticle surfaces have
been presented previously by Budijono [2] and Kumar [3] and are briefly summarized below.
The blob size of the PEG chains in a brush regime, ξbrush, is given as,
3/10
4/34 Bbrush
eff
k TNbξγ π
=
(1)
In Equation S1, kB is the Boltzmann constant, T is the absolute temperature (where T = 298 K is
assumed), N is the number of monomers in the chain, b is the monomer length, and γeff is the
effective interfacial tension at the polystyrene (PS)/PEG/water interface. The effective interfacial
tension is expressed as a function the blob size (Equation S2), necessitating the use of iteration to
solve for either quantity. The interfacial tension between two surfaces (Equation S3) can be
estimated by the harmonic mean of the polar (γp) and dispersive (γd) surface tension parameters
of the individual materials (Equation S 4) [4].
, ,
2 2
2 21γ γ γξ ξ
= + −
PS PEG PS watereff
brush brush
b b (2)
4 4γ γ γ γ
γ γ γγ γ γ γ
= + − −+ +
d d p pi j i j
ij i j d d p pi j i j
(3)
γ γ γ= +d pi i i (4)
The values for the material properties needed in the two equations above are presented in Table
S1. The blob size can be used to determine the number of chains per unit surface area, which is
defined as σbrush.
2
4σπξ
≡ii
(5)
Comment [RKP1]: Carlos, numbe3r these as S1, S2,
Comment [RKP2]: brush or i?
The thickness of the brush layer, δbrush, is given in Equation S6.
2/3 5/3brush brush Nbδ ξ −= (6)
Furthermore, as a comparison, the surface area covered by each chain in a mushroom
conformation can also be estimated. As done by Auguste and coworkers [5], the blob size of the
PEG chains was determined by Equation S7, where MPEG is the molecular weight of the PEG
chain.
[ ]1/20.076 nmmushroom PEGMξ = (7)
From the blob size, the number of chains per surface area for the mushroom regime, σmushroom,
was calculated by using Equation S5. Under the mushroom regime, the thickness of the polymer
layer, δmushroom, is equal to the size, ξmushroom. The theoretical values are listed in Table S2.
Material Properties for Blob Size and Surface Coverage Calculations
b = 3.5 Å [10] Mb = 44 g/mol
γwaterd = 22.1 mN/m [4]
γwaterp = 50.7 mN/m [4]
γPEGd = 30.9 mN/m [3]
γPEGp = 12 mN/m [3]
γPSd = 40.1 mN/m [16]
γPSp = 0.6 mN/m [16]
ρPS = 0.969 g/mL [17] ρβ-carotene = 1.000 g/mL [202] ρα-tocopherol = 0.950 g/mL [202] ρPPG = 1.004 g/mL [202]
Table S1 Material properties of polymers, core solutes, and water required for calculating the
blob size of PEG chains, as well as the experimental surface coverage of NPs. The references
from which each physical property was obtained are given.
Now to compare to the general theory, an iterative method making use of the
experimental data was used to calculate "experimental" PEG properties. For these calculations,
the reported number distributions from the Zetasizer were converted to surface area-weighted
distributions, as the process of particle aggregation is a surface phenomenon. From these
distributions, the surface area moment mean diameter (SMD) was calculated and used as the
particle diameter, DNP. In Equation S8, nd is the number of particles reported in a size class and
d is the diameter of the size class [6].
[ ]3
23, 2 dNP
n dD D
d≡ = ∑
∑ (8)
The size measured by dynamic light scattering is that of the nanoparticle core plus the thickness
of the polymer layer, due to hydrodynamic screening by the PEG layer [7, 8]. Thus, the volume
of a β-carotene/polystyrene core was calculated by subtracting twice the theoretical PEG chain
length from the SMD; regardless of which conformation is assumed for the chain length, the end
result converges to the same value.
Based on the mass of β-carotene and polystyrene from the block copolymer (BCP), the
total volume of core solute can be calculated from material densities. By dividing the total
volume by the volume of a core, the number of particle cores is calculated. When calculating the
number of block copolymer chains present in the system, it can be assumed that all of the block
copolymer is on the particle surfaces because no micelle populations were observed for these
batches, which would be detectable by dynamic light scattering if they did exist [2, 9]. From
these values, the number of chains per particle, or the aggregation number, p, can be determined
and it can then be converted to the number of chains per surface area, σ1, by dividing the chains
per particle by the surface area per particle core. All steps are condensed in the equation
( )1
2
6
BCP Av NP
iBCP
i i
m N DmM
δσ
ρ
−=
∑ (9)
In Equation S9, mBCP is the mass of block copolymer, NAv is Avogadro's number, MBCP is the
molecular weight of the block copolymer, ρi is the density of the individual components in the
core (β-carotene and PS in this case), and mi is the mass of the individual core components.
Finally, based on the surface coverage, the PEG layer thickness, δ, was calculated using the
method outlined by Biver et al, which assumes a brush conformation on a curved surface [10].
Curvature of the particle surface becomes significant for the case of the smallest particles where
the PEG chain length is at 20% of the particle radius. The main equation has been recast below
in terms of the aggregation number, p.
5/3 3/51/32
32 2 4NP NPD D v b pkNb
bδ
π
= − −
(10)
Here, k is a constant O(1) that was taken to be 1 and v/b3 is the excluded volume parameter,
which was taken as 1.75 for PEG in water [11]. The layer thickness (Equation S10) is used
iteratively to obtain a self-consistent solution for δ and p: the new δ is used to recalculate the
volume of a particle core, the number of particles, the aggregation number, and subsequently a
new layer thickness. This process is repeated until the layer thickness converges, from which the
experimentally determined surface coverage can be determined using Equation S 11.
2
expexp
exp
214 / 2NP
pD
σπ δ
− ≡ −
(11)
The final experimental values, as compared to the theoretical values for the blob size,
layer thickness, and surface coverage, are listed in Table S2.
Sample ξ (nm) δ (nm) σ (chain/nm2) pexp
(chains/particle) exp brush exp mush brush exp mush brush
PEG1.8K: 12/8 1.56 1.02 3.90 3.22 7.01 0.52 0.12 1.22 1008
PEG1.8K: 10/10 1.73 1.02 3.69 3.22 7.01 0.43 0.12 1.22 929
PEG1.8K: 7/13 0.98 1.02 5.48 3.22 7.01 1.32 0.12 1.22 47641
PEG1.8K: 5/15 1.08 1.02 5.19 3.22 7.01 1.09 0.12 1.22 59822
PEG1.8K: 3/17 1.35 1.02 4.50 3.22 7.01 0.70 0.12 1.22 49576
PEG2.5K: 12/8 2.16 1.12 4.30 3.80 9.14 0.27 0.09 1.01 267
PEG2.5K: 10/10 1.44 1.12 5.68 3.80 9.14 0.62 0.09 1.01 3006
PEG2.5K: 7/13 1.31 1.12 6.21 3.80 9.14 0.75 0.09 1.01 12458
PEG2.5K: 5/15 1.41 1.12 5.97 3.80 9.14 0.64 0.09 1.01 17677
PEG2.5K: 3/17 1.48 1.12 5.85 3.80 9.14 0.58 0.09 1.01 40276
PEG5.3K: 12/8 2.10 1.39 8.58 5.53 16.78 0.29 0.04 0.66 547
PEG5.3K: 10/10 1.65 1.39 10.51 5.53 16.78 0.47 0.04 0.66 3367
PEG5.3K: 7/13 1.87 1.39 9.97 5.53 16.78 0.36 0.04 0.66 4336
PEG5.3K: 5/15 1.87 1.39 10.21 5.53 16.78 0.37 0.04 0.66 9638
PEG5.3K: 3/17 1.99 1.39 9.99 5.53 16.78 0.32 0.04 0.66 20598
Table S2 Characteristics of hydrated PEG chains of different molecular weights in the
experimental nanoparticle system, as well as in the mushroom and brush conformations. Sample
name codes are as follows. All block copolymers were PS-b-PEG, with a 1.6K PS block (except
for the 1.5K-5.3K) and thus only the molecular weight of the PEG block is listed, followed by
the mass ratio of BCP to β-carotene in the NP formulation: 12/8, 10/10, 7/13, 5/15, 3/17.
VAN DER WAALS ATTRACTION
The van der Waals attractive forces between nanoparticle cores is calculated using the
non-retarded force expression of Hamaker [12], where Fvdw is the attractive force between two β-
carotene cores of diameter DNP separated by a distance x and A is the Hamaker constant for the
suspended nanoparticle system.
( ) ( )( ) ( )2 32 2
2 1 1 2 1 ; 6 2 1 12
vdwNP NP
xA x xF x xD x x x Dxx x
+ + = − − − = + + ++
(12)
The attractive force is integrated from an initial point where two particles have the PEG coronas
touching (2δexp) to the point where the PEG layers have been condensed and the cores are
separated only by a dehydrated PEG layer (2ℓ) (Figure 7 in main paper). The final work
expression is,
( )exp
2
2vdw vdwW F x dxδ
= ∫
(13)
To estimate the Hamaker constant, the Lifshiftz theory is used in the non-retarded regime for a
symmetric case of two identical phases "1" interacting across a medium "3" [13], where εi is the
dielectric constant of phase i, ni is the refractive index of phase i, h is Planck's constant, and ve is
the main electronic UV absorption frequency around 3 x 1015 s-1.
( )( )
22 2 21 31 3
3/22 21 3 1 3
334 16 2
eB
n nhvA k Tn n
ε εε ε
− −= + + +
(14)
Phase "1" is assumed to be pure β-carotene (ε1 = 2.5 [14], n1 = 1.47 [15]) and phase "3" is taken
as ethylene glycol at 15 wt% in water at 20°C (ε1 = 76.4, n1 = 1.35 [201]). The ethylene glycol
solution of 15 wt% is chosen because the stabilizing PEG chains are in solution between the two
cores at an average corona volume fraction, ϕ, of 15% (average for all batches). This estimate
yields a Hamaker constant of 6.64 x 10-21 J.
Symbols Used
ξ blob size of a polymer chain
kB Boltzmann constant T absolute temperature N number of monomers in a polymer chain b monomer length γ surface or interfacial tension σ surface coverage, or the number of polymer chains per surface area, on a surface δ polymer layer thickness M molecular weight ρ density of a material DNP diameter of a nanoparticle n number of particles reported in a size class of diameter d d diameter of a size class m mass of a material NAv Avogadro's number k a constant v/b3 excluded volume parameter of a polymer p aggregation number F force A Hamaker constant x separation distance W work ℓ thickness of a dehydrated polymer layer ε dielectric constant h Planck's constant ve main electronic absorption frequency in the UV range n refractive index ϕ volume fraction
Example Calculation for NP Batch A-1
NP formulation details PS-PEG (1.6K-1.8K)/β-carotene • mPS-PEG = 1.2 mg = 0.0012 g • mPS = 1.6 (1.2 mg) / (1.6 + 1.8) = 0.565 mg = 0.000565 g • mβ-carotene = 0.8 mg = 0.0008 g • VPS = mPS /ρPS = (0.000565 g) / (0.969 g/mL) = 0.00058 mL • Vβ-carotene = mβ-carotene / ρβ-carotene = (0.0008 g) / (1 g/mL) = 0.0008 mL • Vcore = VPS + Vβ-carotene = (0.00058 mL) + (0.0008 mL) = 0.00138 mL • DI = 61.73 nm (intensity-weighted average diameter) • DNP = 31.02 nm Initial surface density • ξmushroom = δmushroom = 0.076 (MPEG)1/2 = 3.22 nm • DNP,core,0 = DNP - 2 δmushroom = (31.02 nm) - 2 (3.22 nm) = 24.58 nm • VNP,core,0 = 4π (DNP,core / 2)3 / 3 = 4π [(24.58 nm) / 2]3 / 3 = 7.77 x 10-18 mL/NP • nNP,core,0 = Vcore / VNP,core = (0.00138 mL) / (7.77 x 10-18 mL/NP) = 1.78 x 1014 NP • nBCP,0 = mPS-PEGNAv / MPS-PEG = (0.0012 g)(6.022 x 1023 chain/mol) / (3400 g/mol) = 2.13 x
1017 BCP chains • p = nBCP / nNP,core = (2.13 x 1017 BCP chains) / (1.78 x 1014 NP) = 1194.54 chains/NP • SANP,core = 4π (DNP,core / 2)2 = 4π [(24.58 nm) / 2]2 = 1897.36 nm2/NP • σ1 = p / SANP,core = (1194.54 chain/NP) / (1897.36 nm2/NP) = 0.63 chains/nm2
• ( )1
2
6
BCP Av NP
iBCP
i i
m N DmM
δσ
ρ
−=
∑ = (0.0012 g)(6.022 x 1023 chain/mol)[(31.02 nm) - 2 (3.22 nm)] /
6 (3400 g/mol)[(0.000565 g) / (0.969 g/mL) + (0.0008 g) / (1 g/mL)] (mL/cm) (cm / 107 nm)3 = 0.63 chains/nm2
•
5/3 3/51/32
1 3 4.16 nm2 2 4NP NPD D v b pkNb
bδ
π
= − − =
Iteration Iterate the following steps until the chain length converges:
− DNP,core,i = DNP - 2 δi − VNP,core,i = 4π (DNP,core,i / 2)3 / 3 − nNP,core,i = Vcore / VNP,core,i − pi = nBCP / nNP,core,i
− 5/3 3/51/32
32 2 4NP NP i
iD D b pvkNb
bδ
π
= − −
Final values: − pexp = 1008 chains/NP − δexp = 3.90 nm
− ( )51exp
expexp
4 4 3.90 nm1.56 nm
2 1008 2pδ
ξ = = =− −
− ( )
2exp 22
exp
4 4 0.52 chain/nm1.56 nm
σπξ π
= = =
Repulsive PEG osmotic pressure work • VNP,PEG = pf (MPEG / NAv) / ρPEG = (1008 chain/NP) [(1800 g/mol) / (6.022 x 1023 chain/mol)] /
(1.046 g/mL) = 2.88 x 10-18 mL/NP • DNP,condensed = 2[(VNP,PEG + VNP,core) / (4π/3)]1/3 = 2[(2.88 x 10-18 mL/NP + 7.77 x 10-18
mL/NP) / (4π/3) (mL/cm)]1/3 (cm / 107 nm) = 26.22 nm • ℓ = (DNP,condensed - DNP,core) / 2 = 26.22 nm) - [(31.02 nm) - 2 (3.90 nm)] / 2 = 1.50 nm
• ( ) ( )( )( ) ( )( )
9/223 9/4 3/4
3
1.38 10 J/K 195 K 3.90 nm 3.90 nm0.80.35 nm 40.9 3.90 nm0.35 nm
xxx
−× Π = −
• ( ) ( ) ( )2 2
1.50 19
3.90
31.02 nm 31.02 nm 3.90 nm 1.28 10 J4 2stericW x x dxπ −
= Π − − + = − × ∫
Attractive van der Waals work
• ( ) ( )( )
( )( ) ( )
21
2 32 2
6.64 10 J 2 1 1 2 1 ; 6 31.02 nm 2 1 31.02 nm12
vdw
x x xF x xx x x xx x
− × + + = − − − = + + ++
• ( )
( )
( )2 1.50 21
2 3.901.50 10 Jvdw vdwW F x dx −= = ×∫
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