Abstract
We employed a combined experimental and analytical approach to investigate the influence of nanoparticle geometry on the loading efficiency of doxorubicin (DOX) onto 
nanoparticles. Experimentally, three distinct 
shapes, spherical, sheet, and cylindrical, were synthesized, characterized, and their respective DOX loading efficiencies were measured. Concurrently, analytical mathematical models were developed to calculate the van der Waals (vdW) interaction energy between a spherical DOX molecule and each nanoparticle geometry, considering both theoretical loading and surface adsorption scenarios. The model successfully predicted the relative thermodynamic stability by yielding high and similar binding energies for the spherical and sheet geometries, which aligned well with their high experimental loading efficiencies. However, a significant quantitative discrepancy arose with the cylindrical shape, where the predicted binding energy did not correspond to the high experimental loading efficiency. This divergence powerfully demonstrates that a simple vacuum-based vdW model is fundamentally insufficient to fully capture the complexity of the drug-nanoparticle interaction. Despite this limitation, the synergy between experimental validation and theoretical modeling provides a critical framework for understanding the geometric dependence of drug-nanoparticle interactions and guides future model refinement toward incorporating the complexity of the nano-bio interface.
Subject terms: Materials science, Nanoscience and technology, Physics
Introduction
Nanostructures have been crucial in advancing drug and medical research for decades, establishing the field of nanomedicine. The use of nanoscale materials has led to a wide range of applications in the diagnosis and treatment of various diseases, including kidney1, cardiovascular2, and cancer-related diseases3,4. Furthermore, several types of nanoparticles have demonstrated the properties such that enhance the efficiency of drug delivery systems in reaching target cells such as liposomes, polymeric and solid lipid nanoparticles5–13. Cerium oxide (
) nanoparticle is also considered as one of the effective carriers in drug delivery, showing remarkable antioxidant properties, antibacterial activity, excellent biocompatibility and the capacity to support wound healing14–16. As a result, these properties lead to a powerful potential of 
in cancer therapy by minimizing toxicity and maximizing killing ability for anticancer drug13,17,18.
A severe increase in the global incidence of cancer has necessitated the development of numerous advanced therapeutic treatments and pharmacological agents. Doxorubicin (DOX) can be administered more effectively to treat many types of cancer when loaded with nanoparticles. For example, attaching 
nanoparticles with DOX increases the ability of DOX to fight breast cancer, making them promising new treatments19.
Recent advances in nanomedicine have demonstrated that nanoparticle geometry plays a pivotal role in optimizing doxorubicin (DOX) drug loading and delivery. The shape, size, and surface features of nanocarriers such as nonionic surfactant vesicles20, black phosphorus nanoparticles21, and DNA origami structures22 greatly influence DOX encapsulation efficiency, release kinetics, and therapeutic performance. These innovative geometries enable improved targeting, controlled drug release, and enhanced cytotoxicity, emphasizing the importance of geometric tuning and surface engineering for more effective and safer DOX-based cancer therapies. The shape and size of nanoparticles are also vital for their effectiveness23 so many conformations are studied such as spheres, rods, disks, and cones. The size impact on factors relating to their circulation duration, targeting efficiency, and cellular uptake. In addition, different shapes can influence how cells absorb them and can improve the precision of drug delivery24,25. For instant, spherical nanoparticles produce the most favorable outcome in terms of drug adsorption24 and non-spherical particles such as rods and wormlike particles, possess a higher drug loading capacity25,26. Therefore, it is essential to also consider the shape of the nanoparticles to optimize the efficiency of the delivery system.
Recent progress in theoretical modeling has profoundly enhanced our understanding of nanomaterials, primarily through the integration of advanced van der Waals (vdW) corrected Density Functional Theory (DFT) methods27,28. These developments include automated algorithms capable of constructing and screening large numbers of bilayer vdW supercells. This capability provides deep insights into how factors like supercell geometry, twist angle, and strain impact electronic and mechanical properties28, thereby overcoming limitations inherent to conventional DFT when modeling metastable configurations in layered vdW heterostructures. Furthermore, recent reviews underscore the necessity of incorporating long-range dispersion and nonlocal interactions for realistic simulation of nanomaterial architectures, particularly in applications like catalysis, energy devices, and environmental technologies27. This necessity highlights that London dispersion interactions are crucial for accurate molecular modeling; explicit consideration of these long-range electron correlations demonstrably improves predictions of molecular stability, reactivity, and material properties. The robust integration of computational and experimental techniques has firmly established dispersion forces as a critical factor in the rational design and comprehensive understanding of modern nanomaterials29.
Mathematical modeling is another key in developing nanostructure geometries, especially for predicting drug delivery system30. The Lennard-Jones potential function is adopted as a tool in finding the interaction energy between two non-bonded atoms which is beneficial for the finding of energy optimization. Several studies use the model to describe the encapsulation capacity of multiple conformations of DOX including sphere, cylinder, and ellipse, for various carriers such as liposome, peptide nanotube, and lipid nanotube31–33. Consequently, to improve the efficiency in using 
as a carrier delivering drug DOX to targeted cells, we investigate the interaction energy, using the Lennard-Jones potential function, between three conformations of 
, which are sphere, planar sheet and cylinder, and a spherical drug DOX.
The current understanding of drug-nanoparticle interactions often relies on either experimental observations or computational simulations. However, a comprehensive study that systematically links a fundamental analytical model with direct experimental validation across various nanoparticle geometries remains largely unexplored. This study addresses this gap by presenting a combined approach to investigate the interaction between a DOX molecule and three distinct 
nanoparticle shapes. Analytical expressions for the vdW energy are derived for spherical, sheet, and cylindrical geometries and are directly compared with experimental measurements of DOX loading efficiency. This integrated methodology provides a unique synergy, the analytical models offer a theoretical framework that explains the observed experimental trends, while the experimental data validates the models’ predictive capabilities. This approach provides a deeper and more complete understanding of how nanoparticle geometry fundamentally influences drug-nanoparticle interactions.
The following sections provide a detailed overview of this study. The experimental details for the synthesis and characterization of different shapes of 
nanoparticles, along with the measurement of DOX loading efficiency, are described in Section 2. Section 3 presents the mathematical derivation for the non-bonded interaction energy of the molecules. The numerical results from the analytical expressions, including a comparison with our experimental findings, are provided in Section 4. Finally, a summary of the key findings is given in Section 5.
Experimental study
We begin by synthesizing 
nanoparticles in different shapes and characterizing their morphology using Transmission Electron Microscopy (TEM). Following characterization, we determine the loading efficiency of DOX onto each 
shape and evaluate the cytotoxicity on breast cancer cells.
Materials
Cerium (III) nitrate hexahydrate; Ce(
)
·6
O 99% (Sigma Aldrich), sodium hydroxide anhydrous; NaOH 
(Carlo Erba), ethylene glycol; 
OH 
(Sigma Aldrich), ethyl alcohol; 
OH 99.9% (Duksan), acetic acid; 
COOH 
(Carlo Erba); dimethyl sulfoxide (DMSO) and 3-(4,5-dimethylthiazol-2-yl)-2,5-diphenyltetrazolium bromide (MTT) (Sigma-Aldrich); Doxorubicin hydrochloride (Thermo Fisher Scientific); Dulbecco’s Modified Eagle Medium (DMEM) (Thermo Fisher Scientific); Penicillin-streptomycin (Pen-strep, P/S) (Gibco).
Synthesis of different shapes of 
Synthesis 
nanosphere (C-sphere)
nanosphere were synthesized based on the previous protocol with minor modification34. Briefly, 2.8092 g of Ce(
)
6
O was dispersed in a mixed solvent consisting of water (2.8 mL), acetic acid (2.8 mL), and ethylene glycol (70 mL). The solution was stirred using a magnetic stirrer for 30 minutes to ensure homogeneity. The resulting mixture was transferred into a stainless-steel autoclave and subjected to a solvothermal reaction at 
C for 200 minutes. After cooling to room temperature, the precipitate was collected via centrifugation at 19,319 
g for 10 minutes at 
C. The pellet was washed five times with 35 mL of deionized water until the pH reached pure deionized water, followed by three washes with 35 mL of ethanol. The resulting solid was dried in a hot air oven at 
C for 12 hours and then calcined at 
C for 4 hours in a muffle furnace (a heating rate of 
C/min). The final product was weighed, and the yield was calculated.
Synthesis 
nanorods (C-cylinder)
nanorods were synthesized following previously reported methods with minor modifications35. 0.8680 g of Ce(
)
6
O was dissolved in 30 mL of 6 M NaOH under magnetic stirring for 30 minutes, resulting in a milky white suspension. The mixture was transferred into a stainless steel autoclave and subjected to a solvothermal treatment at 
C for 24 hours. After the reaction, the suspension was centrifuged at 19,319 
g for 15 minutes at 
C. The resulting precipitate was washed five times with 35 mL of deionized water until neutral pH (
), followed by three ethanol washes (35 mL each). The obtained solid was then dried at 
C for 24 hours and subsequently calcined at 
C for 4 hours. The product was weighed to determined yield.
Synthesis 
sheet-shaped (C-sheet)
nanosheets were synthesize following previously established procedures with minor modifications36. Ce(
)
6
O was dissolved in ultrapure water and heated at 
C on a hot plate with magnetic stirring for 20 minutes. The pH of the solution was adjusted to 12 using NaOH. The mixture was stirred continuously stirrer at 
C for 24 hours. The resulting nanoparticles were collected via centrifugation at 15,090 
g for 30 minutes and washed with deionized water and ethanol until the supernatant reached neutral pH (
). The nanoparticles were then freezing dried for 25 hours, and subsequently calcined at 
C for 2 hours in a muffle furnace. The sheet-shaped cerium oxide nanoparticles were collected and stored for further use.
Transmission electron microscopy (TEM)
TEM images were obtained by JEOL model JEM-2010 TEM operating at 120-200 kV.
Dynamic light scattering analysis (DLS)
The hydrodynamic diameter of the synthesized nanoparticles was measured using a Zetasizer Ultra (Malvern Panalytical) based on the principles of DLS. All measurements were performed at a controlled temperature of 
C. Prior to analysis, nanoparticle suspensions were diluted 1:100 in deionized water to a final concentration of 0.1 mg/mL, ensuring minimal multiple scattering and optimal accuracy.
Morphological characterization of 
nanoparticles were synthesized using a hydrothermal method to obtain distinct morphologies. TEM was employed to characterize the spherical, cylindrical, and sheet shaped structures of 
, as shown in Fig. 1. The spherical 
nanoparticles (C-sphere, Fig. 1(a)) exhibited uniform size distribution with well-defined geometry, and quantitative analysis revealed an average diameter of 
nm. The cylinder-shaped 
nanoparticles (C-cylinder, Fig. 1(b)) appeared as elongated particles often forming aggregated networks, with an average length of 
nm and width of 
nm, corresponding to an aspect ratio of 
. The sheet-like 
(C-sheet, Fig. 1(c)) displayed a transparent, thin, and layered morphology with partial aggregation into overlapping structures, consistent with their two-dimensional nature; size analysis showed an average lateral dimension of 
nm and width of 
nm. The total number of measured particles (
) represents the overall morphology and size uniformity across samples. The averaged size are reported as mean ± SEM. Additionally, the TEM size distribution histograms for all three 
nanoparticle morphologies are provided in Supplementary Figure S1. The size distribution histograms further confirmed consistent particle formation with minor variations among morphologies. These results confirm successful synthesis of 
nanoparticles with morphology-controlled structures, which may critically influence their physicochemical properties and subsequent biological or catalytic activities.
Fig. 1.
The microscope image from TEM technique includes (a) C-sphere with scale bar = 200 nm, (b) C-cylinder, and (c) C-sheet with scale bar = 20 nm.
DOX loaded on different shapes of 
Three different shapes of 
nanoparticles were dispersed in PBS at a concentration of 1 mg/mL by stirring for 15 minutes. Separately, 500 
M of DOX was was added to 200 
L of 1 mg/mL 
nanoparticles in 1
PBS, and the mixture was vortexed. The two solutions were combined and stirred in the dark for 
hours. After incubation, the mixture was subjected to ultracentrifugation at 15,000 
g for 10 minutes using a Multifuge X Pro Series ultracentrifuge (Thermo Scientific). The resulting cell pellet, representing the DOX-loaded 
nanoparticles, was collected, dried at 
C, and stored at room temperature. All experiments were performed in triplicate (
) to ensure reproducibility. The supernatant was collected and analysed for loading efficiency (%LE) using a UV-Vis Spectrophotometer (Shimadzu) over a wavelength range of 
nm. The blank correction procedure, performed by subtracting the absorbance of the corresponding solvent control. The determination is based on quantifying the unabsorbed DOX in the supernatant at an absorbance maximum of 480 nm, according to the equations:
 |
The optical properties of 
nanoparticles and their DOX-loaded nanoparticles (
NPs–DOX) were investigated to assess morphology-dependent drug-loading behavior. In Fig. 2(a), UV–vis spectra of bare 
nanoparticles showed a characteristic absorption peak around 320 nm, with no observable DOX absorbance at 480 nm, indicating the absence of free drug across all nanoparticle shapes.
Fig. 2.
UV–vis absorption and fluorescence spectral analysis of 
and 
NPs-DOX with different morphologies. (a) UV–vis of bare 
, (b) UV–vis and (c) fluorescence emission spectra of supernatants after incubation with DOX for different 
morphologies. The absorbance corresponds to unbound DOX, decreased peak intensity indicates greater drug uptake by the nanoparticles. (d) Visual confirmation of DOX loading under UV illumination. DOX emitted fluorescence, confirming the successful loading of DOX onto the 
.
To evaluate %LE, the supernatant collected after DOX loading was analyzed. Representative UV–Vis spectra of DOX solutions and the corresponding blank are provided in Supplementary Figure S2. UV–Vis spectra of the supernatants clearly showed the characteristic DOX absorbance peak, and decreasing intensity across different nanoparticle morphologies indicated varying amounts of drug uptake as shown in Fig. 2(b). These results confirmed the successful loading of DOX into all three nanoparticle morphologies. The UV–Vis calibration curve of DOX was constructed, showing excellent linearity (
). Limit of detection (LOD) and limit of quantification (LOQ) for DOX were calculated from the calibration line using the equations LOD = 
SD/slope and LOQ = 
SD/slope, as previously reported37. Based on the standard deviation of the blank and the slope of the calibration curve, the LOD and LOQ were determined to be 2.83 and 8.57, respectively. The %LE were calculated from the difference between the initial and remaining DOX concentrations. The loading efficiency values are reported as mean ± SEM from three independent experiments (
). Among the various morphologies of 
nanoparticles, C-sphere-DOX exhibited the highest loading efficiency (
), followed by C-cylinder-DOX (
) and C-sheet-DOX (
) (see Table 1). The hydrodynamic sizes of 
NPs-DOX were measured using DLS and are summarized in Table 1. The percent loading efficiency of C-sphere-DOX and C-cylinder-DOX was significantly higher than that of C-sheet-DOX (
and 
, respectively), as determined by one-way ANOVA followed by Tukey’s multiple comparison test.
Table 1.
Characterization of cerium oxide nanoparticles with and without doxorubicin. Data are presented as mean ± SEM from three independent experiments (
).
| Sample | Size (nm) | PdI | Zeta potential (mV) | Loading efficiency (%) |
|---|---|---|---|---|
C-
|
 |
 |
 |
− |
C-
|
 |
 |
 |
− |
C-
|
 |
 |
 |
− |
C-sphere-
|
 |
 |
 |
 |
C-cylinder-
|
 |
 |
 |
 |
C-sheet-
|
 |
 |
 |
 |
NPs were dispersed in 
water solution, 
PBS solution. Statistical analysis was performed using one-way ANOVA followed by Tukey’s multiple comparison test. 
(C-sphere-DOX vs C-sheet-DOX); 
(C-cylinder-DOX vs C-sheet-DOX).
Among the different shapes, C-sheet-DOX exhibited the largest particle size, likely due to greater aggregation tendencies in aqueous solution, followed by C-cylinder-DOX, whereas C-sphere-DOX showed the smallest size. DLS measurements were conducted in water and in PBS. A fluorescence spectrum of unbound DOX in supernatant of DOX-loaded 
nanoparticles further supported drug loading: C-sphere–DOX showed the lowest DOX fluorescence intensity, indicating minimal unbound drug and effective loading as shown in Fig. 2(c).
The visual confirmation of drug loading was obtained under UV illumination as presented in Fig. 2(d). While bare C-sphere suspension showed negligible fluorescence (left), C-sphere-DOX suspension exhibited a clear fluorescence detectable by the naked eye, confirming successful DOX loading onto 
particles (right). Overall, these results demonstrate that the shapes of 
nanoparticles significantly affects their optical behavior and drug-loading performance of 
nanoparticles based delivery systems.
In vitro evaluation of anticancer effects of DOX loaded 
on breast cancer cells
The cytotoxicity of free DOX, bare 
, and DOX-loaded 
nanoparticles with different shapes was evaluated in MDA-MB-231 breast cancer cells using the MTT assay. The percent cell viability of each bare 
morphology and DOX-loaded 
was compared in Fig. 3 using the same concentration of nanoparticles, while the cytotoxicity profile of free DOX was provided in the Supplementary Figure S3. Among the bare nanoparticles, C-sphere exhibited the highest cytotoxicity with an 
of 
g/mL, whereas C-cylinder and C-sheet 
did not reach 
within the tested concentration range, indicating low toxicity. DOX-loaded 
exhibited greater cytotoxicity than bare 
. Among the shapes tested, C-sphere-DOX exhibited the lowest 
(
g/mL), followed by C-cylinder-DOX (
g/mL) and C-sheet-DOX (
g/mL) as shown in Table 2, indicating superior drug delivery efficiency. Notably, the potent cytotoxicity of C-sphere-DOX correlated with its highest loading efficiency, suggesting that nanoparticle shape plays a critical role in drug loading and therapeutic efficacy, with C-sphere-DOX being the most effective formulation for breast cancer treatment.
Fig. 3.
The cytotoxicity of bare 
compared to DOX-loaded 
NPs on MDA-MB-231 cells. (a) C-sphere and C-sphere-DOX, (b) C-cylinder and C-cylinder-DOX, (c) C-sheet and C-sheet-DOX, and (d) All DOX-loaded 
NPs. Data are expressed as mean ± standard error of the mean (
). 
, 
, and 
; Student’s T-test.
Table 2.
The 
of bare 
and DOX-loaded 
with different shapes in MDA-MB-231 cells.
 ( g/mL) |
|||
|---|---|---|---|
| C-sphere | C-cylinder | C-sheet | |
| Bare NPs |  |
N/A | N/A |
| Loading with DOX |  |
 |
 |
N/A (not applicable).
In vitro drug release of DOX was investigated using PBS (pH 7.4) to simulate the internal environment. Briefly, 2 mg of DOX-loaded nanoparticles were suspended in 1 mL of PBS and placed in a dialysis membrane (MW cutoff 
Da) with moderate stirring (120 rpm) at 
C. At predetermined time points, 1 mL of release medium was collected and immediately replaced with fresh PBS. Released DOX was quantified using a spectrofluorometer (Jasco) at an excitation of 493 nm and emission of 590 nm, and release percentage was calculated as
 |
The cumulative release of DOX exhibited a time-dependent profile (Supplementary Figure S4). Free DOX showed a rapid burst (
% at 1 h), gradually reaching 
% at 24 h and plateauing up to 72 h. In contrast, DOX release from 
nanocarriers was slower and morphology-dependent: C-sheet-DOX reached 
% at 72 h, C-cylinder-DOX 
%, and C-sphere-DOX 
%, indicating strong drug retention in compact particles. Loading within 
reduced burst release and prolonged drug release. Interestingly, the trend in release kinetics (C-sheet > C-cylinder > C-sphere) inversely correlated with cytotoxicity (C-sphere > C-cylinder > C-sheet), suggesting that slower release from highly loaded, compact nanoparticles promotes prolonged intracellular retention of DOX, enhancing cell killing despite lower cumulative release.
Interaction energy between DOX and 
In order to determine the capacity of delivering DOX to targeted cell, we assume that DOX is modeled as a sphere of radius a interacting with the three conformations of 
surfaces including spherical shape, planar sheet, and cylindrical shape. The Lennard-Jones potential function is utilized to evaluate the interaction energy between two non-bonded atoms, and it is given by
 |
1 |
where 
and 
are mean atomic densities of two structures, 
is a distance between atoms and 
and 
are the attractive and repulsive constants, respectively. The values of 
is the well-depth and 
is the van der Waals diameter of surface 
and 
and they are taken from the work of Rappe et al.38.
The Lennard-Jones constants A and B in the continuous approach may be determined by the weighted average proportional to the atomic types of the two interacting molecules. For example, the constants for the interaction between DOX (
) and 
sheet are given by
 |
where DOX has a total of 68 atoms in a molecule. The fractions 1/3 and 2/3 represent the percentage mixing between cerium and oxygen in 
.
We first determine vdW energy between a spherical molecule and an atom as depicted in Fig. 4. For more convenience, we define
 |
2 |
where 
and 
, so the interaction energy of a sphere interacting with a point becomes 
.
Fig. 4.
Schematic model for atom located at 
interacting with sphere of radius a centered at origin.
The spherical DOX of radius a is assumed to be centered at the origin with its surface element 
, and the atom is assumed to be located at 
. Then the square of the Euclidean distance between the surface of the DOX and the atom is given by
 |
Therefore, 
defined by (2) becomes
 |
By substituting 
, we have
 |
3 |
For 
and 
, we may deduce
 |
Next the atom will be assumed to be an arbitrary point on the surface of 
nanoparticles. Consequently, a second surface integration is necessary to calculate the total interaction energy of the system. Our goal is to determine the distance between the DOX molecule and the 
structures and then perform another surface integral. Specifically, we rewrite 
and 
as a difference of squares, 
. The expressions for these are
 |
4 |
 |
5 |
To further extend this analysis, we introduce another integral 
defined by
 |
6 |
where m is an integer exponent in equations (5) and (4). Consequently, equations (5) and (4) can be expressed in the form
 |
For the full, step-by-step derivations and detailed integral manipulations necessary to reach the closed-form expressions presented in this section, readers are kindly directed to our published work39.
In the following sections, we will evaluate the interaction energy between the spherical DOX molecule and three different shapes of 
nanostructures. The schematic models for these are presented in Fig. 5. The 
structures are assumed to be centered at the origin, while the center of the DOX molecule is shifted along the z-direction by a distance d. The parameter 
, as shown in equations (3) and (6), now represents the Euclidean distance from the center of the DOX molecule to a surface element on the 
structure. Our primary objective is to determine the minimum total energy of the system by treating the total energy as a function of the inter-spacing 
, which is defined as the shortest distance between the surface of the spherical DOX and the surface of the 
nanostructure.
Fig. 5.
Schematic models for spherical DOX molecule interacting with (a) spherical, (b) planar and (c) cylindrical shapes of 
nanoparticles.
Spherical DOX interacting with spherical 
We assume the spherical 
nanoparticle has a radius of 
, as shown in Fig. 5(a). Note that when 
, the DOX molecule is located on the surface of the 
nanoparticle, while when 
, the drug molecule is encapsulated inside it. The squared distance between the center of the DOX molecule and a surface element on the 
nanoparticle is given by
 |
The integral 
given in (3) may be deduced
 |
where we define the integral to be 
for the second surface element of another sphere. As there is no dependence on 
in the integrand, this can be done immediately and then we make the substitution of 
for 
, which yields
 |
7 |
Therefore, the total interaction energy between two spherical molecules is given by
 |
8 |
where 
for 
and 6 are defined by (7).
Spherical DOX interacting with planar sheet of 
As illustrated in Fig. 5(b), the squared distance from the center of the DOX molecule to a point on the xy-plane is given by 
. Due to the short-range nature of the vdW force with the cut-off distance 
nm40, the 
sheet can be approximated as an infinite flat plane. Utilizing the integral 
defined in equation (6), we introduce 
to represent the contribution from the second surface element, which is an infinite plane. This allows us to deduce
 |
We then make a change of variable substitution of 
which transforms the integral to
 |
By using an integral definition of the beta function, which is
 |
and 
, yields
 |
We repeat this process for the substitution of y, finally we have
 |
9 |
The analytical expression for the total interaction energy between a sphere of radius a and a planar sheet is
 |
10 |
where 
for integers m are given by (9).
Spherical DOX interacting with cylindrical 
This section evaluates the interaction energy between a sphere and an infinite cylindrical surface with a radius of 
. The infinite body assumption is an acceptable simplification given the short-range nature of the vdW force and is necessary to maintain analytical tractability. A schematic of this model is presented in Fig. 5(c). As with the spherical model, the location of the DOX molecule, whether it is inside or outside the nanoparticle, can be determined by the relationship between the cylinder’s radius 
and the vertical distance d. The squared distance from the center of the sphere to any arbitrary surface element on the cylinder is expressed as
 |
where we utilized the identity 
. Given that vdW forces are short-range, we model the cylinder as an infinite surface and integrate over its entire length y from 
to 
. This simplification is valid because the contribution to the total energy from the cylindrical ends is assumed to be negligibly small.
Based on the expression for the integral 
given in equation (6), we introduce the term 
to represent the contribution from the second surface element, which is an infinite cylinder, and this allows us to deduce
 |
We define 
, and make a substitution 
to obtain
 |
Again, we use the integral definition for beta function, and use the face that 
is an even function in terms of 
, we have
 |
By introducing the variable 
, the expression is transformed into
 |
where 
. The above integral is in the Euler form of the hypergeometric function
 |
and 
is a gamma function. Then we have
 |
11 |
where 
. The final analytical expression for the total interaction energy between a sphere of radius a and an infinite cylinder of radius 
is
 |
12 |
where 
for integers m are given by (11).
Numerical results
In this study, DOX molecule is modeled as a sphere with a radius of 0.75 nm as reported in41. We utilize a mean atomic surface density of 
, an optimal value determined from U-NSGA-III simulations of the DOX-graphene interaction that yielded a calculated energy of 
kcal/mol42. For the 
nanoparticles, we calculate the mean atomic surface density using the formula for tessellated hexagonal rings, given by
 |
where 
represents the bond length. With a 
bond length of 
nm, the mean surface density is calculated to be 
. This value is consistently applied to all three 
nanoparticle structures examined in this study.
To validate our analytical expressions for a spherical DOX interacting with spherical, planar sheet, and cylindrical 
nanoparticles, we consider two distinct scenarios, the DOX molecule is either inside the nanoparticle or on its surface. The specific expressions used for the spherical, sheet, and cylindrical interactions are provided in equations (8), (10), and (12), respectively. We model the spherical and cylindrical 
nanoparticles with a radius of 20 nm, a size large enough to capture the full vdW energy of the system40. The resulting energy profiles as a function of the inter-spacing 
are presented in Fig. 6. In this figure, a negative value of 
(dashed lines) indicates the DOX molecule is inside the nanoparticle, while a positive value (solid lines) shows it is on the surface. The numerical values for the inter-spacing 
at which these energy minima occur are reported in Table 3.
Fig. 6.
Interaction energy profiles for DOX sphere of radius 
nm and various 
nanoparticle geometries. The energy is plotted as a function of inter-particle spacing 
. Dashed lines correspond to DOX being located inside 
(
), while solid lines indicate its position on the surface (
). Both the spherical and cylindrical 
are modeled with radius of 20 nm. (a) Full range of attractive and repulsive forces and (b) zoomed view of the attractive equilibrium.
Table 3.
Minimum interaction energy 
(kcal/mol) and corresponding inter-spacing 
(nm) for spherical DOX (
nm) interacting with various 
nanoparticle shapes (sphere, sheet, and cylinder). Results are presented for local energy minima found both inside and on the surface of the nanoparticles. The spherical and cylindrical nanoparticles are assumed to have a radius of 20 nm.
 NPs |
DOX inside | DOX on surface | ||
|---|---|---|---|---|
 |
 |
 |
 |
|
| Sphere |  |
0.2742 |  |
0.2739 |
| Sheet |  |
0.2740 |  |
0.2740 |
| Cylinder |  |
0.2741 |  |
0.2740 |
When a DOX molecule is encapsulated inside the nanoparticles, the spherical 
configuration results in the lowest minimum energy, indicating the most stable system among the three geometries. Conversely, the planar sheet geometry yields the highest energy at equilibrium. This difference in stability is attributed to the optimal curvature matching between the inner spherical surface of the DOX molecule and the outer spherical 
nanoparticle.
The order of stability changes for DOX molecules adsorbed on the surface of the nanoparticles. In this case, the lowest minimum energy is observed for the interaction with the 
sheet, while the highest energy occurs with the spherical 
. It is important to note that despite these differences, the minimum energy values for all six scenarios are very close, suggesting a similar binding energy behavior across all three geometries.
Based on our experimental loading efficiency data presented in Table 1, the C-sphere-DOX system exhibited the highest efficiency, followed by C-cylinder-DOX and then C-sheet-DOX. This experimental order of stability mirrors the theoretical calculation for the loading of the drug inside the nanoparticles. The strong agreement between the experimental results and the theoretical predictions for loading suggests that the final location of the DOX molecule in our experiment is likely a combination of both encapsulated and surface-adsorbed states.
Furthermore, as shown in Fig. 6, the binding energy is consistently stronger when the drug is encapsulated inside the nanoparticles than when it is on the surface. This suggests that the encapsulated state is more stable and thus advantageous for drug storage and loading. The lower binding energy on the nanoparticle surface, in contrast, is favorable for a sustained drug release mechanism.
For all cases, the equilibrium inter-spacing 
is found to be a similar value of approximately 0.274 nm. A notable symmetric behavior is observed for the planar sheet geometry. Due to its infinite nature, the loading of the DOX molecule (on one side of the sheet) and its absorption on the surface (on the other side) yield identical energy values and equilibrium distances.
Despite the assumption of infinite size for the sheet and cylindrical geometries in our analytical model, we applied these mathematical expressions to validate the corresponding experimental findings. Based on the nanoparticle sizes detailed in Section 2.3, we used a radius of 100 nm for the spherical 
and 3.5 nm for the cylindrical 
. Assuming the DOX molecules are on the nanoparticle surfaces, we plotted the energy as a function of the positive inter-spacing 
as shown in Fig. 7. We observed that the spherical and sheet geometries yielded very close binding energy values at equilibrium, specifically 33.1193 and 33.4585 kcal/mol, respectively. In contrast, the cylindrical shape produced a lower binding energy of 29.4859 kcal/mol. However, this theoretical result for the cylindrical shape differs from the experimentally measured loading efficiency presented in Table 1. This discrepancy may be attributed to a limitation of our current mathematical model, which neglects the influence of the surrounding media and its associated effects on the vdW forces and overall molecular interaction.
Fig. 7.
Energy profiles for DOX of radius 
nm adsorption on surface of spherical, sheet, and cylindrical 
nanoparticles. The adsorption energy is shown as a function of separation distance 
, and the inset provides a zoomed-in view of minimum energy locations.
The use of a vacuum-based model is methodologically justified for predicting the geometric trend. In a relative comparison of the DOX/Sphere, DOX/Sheet, and DOX/Cylinder binding energies, the inherent complexities of the solvent-mediated forces, specifically the common DOX/Solvent and 
/Solvent interaction energy terms are assumed to effectively cancel. This strategic cancellation allows the simpler model to precisely isolate and quantify the pure effect of surface curvature and geometry on the vdW interaction.
Summary
In this study, we employed a combined experimental and analytical approach to investigate the fundamental interaction between a spherical doxorubicin (DOX) molecule and three distinct 
nanoparticle geometries, spherical, planar sheet, and cylindrical. Our primary goal was to understand how nanoparticle shape influences drug loading efficiency.
Analytically, we developed closed-form mathematical models to determine the van der Waals (vdW) interaction energy. The models considered two geometric scenarios—DOX loaded within the nanoparticles and DOX adsorbed on their surface. Our calculations revealed that the spherical 
system yielded the lowest minimum energy for the loading scenario, while the sheet geometry provided the most stable interaction for surface adsorption. However, across all geometries and scenarios, the minimum energy values were remarkably close, indicating a similar initial binding tendency.
In the experimental phase, the three distinct 
shapes were synthesized, characterized, and their respective DOX loading efficiencies and cytotoxicity toward breast cancer cells were measured. Applying our analytical models—which assume an infinite dimension but were corrected using actual TEM-obtained nanoparticle dimensions—a strong alignment between theory and experiment was observed, the model successfully predicted similar and high relative thermodynamic stability for the spherical and sheet geometries.
Crucially, a significant quantitative discrepancy arose with the cylindrical shape, where the model’s predicted low binding energy did not correspond to the exceptionally high experimentally determined loading efficiency. This divergence highlights two key limitations in the current analytical framework, (i) the neglect of the surrounding solvent effects on the vdW interaction, and (ii) the simplifying assumption of a singular, uniform mean atomic surface density across all three 
morphologies, which ignores actual crystallographic variations (e.g., exposed facets).
In summary, the synergy of these two methods provides a rigorous foundation by establishing the vdW model as a reliable tool for predicting the initial geometric binding tendency. However, the observed failure to quantitatively predict the exceptional loading of the cylindrical 
serves as our most important finding, mandating a new research direction. Future computational efforts must move beyond idealized vacuum conditions to incorporate the full complexity of the nano-bio interface, specifically Lifshitz theory for solvent effects and multi-site modeling for aggregation, to achieve true quantitative predictive power for complex nanomedicine systems.
Supplementary Information
Acknowledgements
S.K. was thankful for the Graduate Research Assistantship (RA) scholarship for financial aid to support education and research of a graduate student, Faculty of Graduate Studies, Mahidol University. A.W. was supported by a Postdoctoral fellowship award from Mahidol University. P.L would like to thank the Kasetsart University Research Development Institute (KURDI).
Author contributions
K.K. and D.B. designed the study. S.K., A.W. and K.K. conducted the experiments. W.S., N.T. and P.L. prepared the experimental materials. P.S. and D.B. performed the analytical calculations. All authors wrote, reviewed and edited the manuscript.
Funding
This research project has been supported by Mahidol University (Fundamental Fund: fiscal year 2025 by National Science Research and Innovation Fund (NSRF) (FF-068/2568). Partial support was also provided by a grant from the Center for Scientific Instrumentation and Platform Services, Faculty of Science, Mahidol University.
Data availability
The datasets used and analysed during the current study available from the corresponding author on reasonable request.
Declarations
Competing interests
The authors declare no competing interests.
Footnotes
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Kanlaya Katewongsa and Duangkamon Baowan contributed equally to this work.
Contributor Information
Kanlaya Katewongsa, Email: kanlaya.pra@mahidol.edu.
Duangkamon Baowan, Email: duangkamon.bao@mahidol.ac.th.
Supplementary Information
The online version contains supplementary material available at 10.1038/s41598-026-36893-5.
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Supplementary Materials
Data Availability Statement
The datasets used and analysed during the current study available from the corresponding author on reasonable request.