Self-assembled Monolayers Research Articles

Enhancing Thermal Management of Graphene Devices by Self-Assembled Monolayers

Enhancing Thermal Management of Graphene Devices by Self-Assembled Monolayers

Triphenylamine-Based Y-Shaped Self-Assembled Monolayers for Efficient Tin Perovskite Solar Cells.

Triphenylamine-based Y-shaped organic sensitizers, specifically TPA-MN (1), TPA-CA (2), TPAT-MN (3), and TPAT-CA (4), are synthesized and utilized as p-type self-assembled monolayers (SAMs) for tin-based perovskite solar cells (TPSCs). These SAMs are developed using low-cost starting materials, primarily from triphenylamine (TPA) components. An extensive analysis is conducted to examine the crystalline, morphological, thermal, optical, electrochemical, and optoelectronic characteristics of SAMs 1-4, and the results are compared. A two-step method is employed to successfully develop tin perovskite layers on all four SAM surfaces. The resulting devices demonstrates PCE in the following order: TPAT-CA (8.1%) > TPAT-MN (6.1%) > TPA-MN (5.0%) > TPA-CA (4.2%). The TPAT-CA molecule, which contains a thiophene spacer, performed better than the other three SAMs in terms of rapid hole extraction rate, high hole mobility, and retarded charge recombination. Consequently, SAM TPAT-CA exhibited the highest device performance with excellent stability over time, retaining ≈90% from the beginning values after storage for 3000 h. The innovative Y-shaped SAMs describe in this study, characterized by their simple and efficient design, have the potential to contribute significantly to the advancement of perovskite photovoltaics, particularly in the development of cost-effective TPSC technology.

Application of PACz-Based Self-Assembled Monolayer Materials in Efficient Perovskite Solar Cells.

Due to the advantages of low interface resistance, high work function, and high stability, PACz family materials have developed rapidly in p-i-n structure perovskite solar cells (PSCs) in recent years. Numerous studies have shown that PSCs prepared on the basis of PACz family materials or their derivatives as hole transport layers (HTLs) generally exhibit superior performance compared to PSCs prepared on the basis of organic HTL PTAA and inorganic HTL NiOx, especially in terms of stability, demonstrating unparalleled charm. Since the application of PACz-like molecule V1036 as a HTL in PSCs in 2018, research reports on high-performance PSCs based on the PACz family HTL have been widely disseminated. Currently, PSCs based on the PACz HTL exhibit a world record value of 26.7% power conversion efficiency (PCE), breaking the long-standing world record held by n-i-p-structured PSCs, indicating its great potential for application in PSCs. The main problem with using PACz as a HTL is that it exhibits poor wettability toward perovskite precursor solutions, is sensitive to the thickness of the HTL and the roughness of the substrate, has poor thermal stability, and lacks preparation methods, making it difficult to achieve large-area device fabrication. This review summarizes the outstanding achievements of PACz to date, describes the mechanism and unique properties of PACz in PSCs, and looks forward to the future development prospects of PACz.

Cooperative Use of N-Heterocyclic Carbenes and Thiols on a Silver Surface: A Synergetic Approach to Surface Modification.

Surface modification through the formation of a self-assembled monolayer (SAM) can effectively engineer the physicochemical properties of the surface/material. However, the precise design of multifunctional SAMs at the molecular level is still a major challenge. Here, we jointly use N-heterocyclic carbenes (NHCs) and thiols to form multifunctional hetero-SAM systems that demonstrate excellent chemical stability, electrical conductivity, and, in silico, catalytic activity. This synergistic effect is facilitated by the high surface mobility and electron-rich nature of NHCs, combined with the strong binding strength of thiols. Scanning tunneling microscopy, electrical conductivity, and scanning electron microscope measurements, as well as density functional theory calculations, were employed to explore the synergistic interactions in the supramolecular SAMs. The van der Waals integration of ballbot-type NHCs and thiols enables the SAMs to exhibit both superior surface anticorrosion properties (attributing to the shift in the d-band center) and low surface resistance originating from the band alignment. Moreover, we find that the deposition sequence of flat-lying NHCs and thiols results in SAMs with different configurations, which can further tune the mechanistic pathway in silico in the acetylene hydrogenation process. Our results provide essential molecular insights into the local electronic control of the new SAM/metal interface and the high stability of the emergent multifunctionality (NHC/thiol)-SAMs forming self-assembled lamellae structures in the nanometer regime.

Identifying the Role of Interfacial Long-Range Order in Regulating the Solid Electrolyte Interphase in Lithium Metal Batteries.

The self-assembled monolayer (SAM) technique, known for its customizable molecular segments and active end groups, is widely recognized as a powerful tool for regulating the interfacial properties of high-energy-density lithium metal batteries. However, it remains unclear how the degree of long-range order in SAMs affects the solid electrolyte interphase (SEI). In this study, we precisely controlled the hydrolysis of silanes to construct monolayers with varying degrees of long-range order and investigated their effects on the SEI nanostructure and lithium anode performance. The results indicate that the degree of long-range order in SAMs significantly influences the decomposition kinetics of the carbon-fluorine bond in lithium bis(trifluoromethanesulfonyl)imide (LiTFSI), promoting the formation of a LiF-rich SEI and profoundly affecting the long-term stability of the highly sensitive anode during electrochemical processes. These findings provide new insights and directions for the molecular design of SAMs tailored for long-lasting lithium metal interfaces.

PH Dependence of Dihydroxyacetone Oxidation by Electrocatalytic Viologen Self-Assembled Monolayers on Gold Electrodes.

Carbohydrate fuel cells garner much research interest as the world's focus shifts from fossil fuels to renewable energy. Many catalyst options are available for carbohydrate fuel cell development, including enzymes and microbes, various metal-based catalysts, and natural or synthetic mediators. Research challenges include low power output, system fouling and poisoning, inefficient electron release, and complex mechanisms, with multiple pathways leading to low product selectivity. Here, we further investigate a novel approach to catalyze carbohydrate oxidation using Au electrodes with viologen self-assembled monolayers (SAMs). SAM-mediated fuel cells have the potential to address the challenges of other catalyst systems by protecting the electrode surface and controlling the local concentration and structure to increase current generation. The effects of increasing pH on dihydroxyacetone (DHA) oxidation by three viologen SAMs on Au electrodes are presented. Current and power generated during DHA oxidation at varying pH were measured and compared to those of bare Au performance. Two of the SAMs produced more current and power than bare Au at elevated pH. The SAM system produced more current and peak power per molecule than both dilute and concentrated homogeneous viologen systems in the same cell setup. These results demonstrate the benefits and limitations of electrodes modified with redox-active groups for the production of electricity from simple sugars and other carbohydrate sources. These results are encouraging for the development of new strategies for electrical power generation from renewable resources.

A Polymeric Hole Transporter with Dual-Interfacial Interactions Enables 25%-Efficiency Blade-Coated Perovskite Solar Cells.

Self-assembly monolayer (SAM) hole transporters, consisting of anchoring, spacer, and terminal groups, have played a significant role in the development of inverted perovskite solar cells (PSCs). However, the weak interaction between perovskite and hydrophobic terminal group of SAMs limits surface wettability and interface stability. To address this issue, two novel hole transporters (named DBPP and Poly-DBPP) with centrosymmetric biphosphonic acid groups are developed. Unlike conventional SAM hole transporters, the biphosphonic acid groups in DBPP and Poly-DBPP can anchor to the underlying conductive substrate and interact with the perovskite layer simultaneously, improving surface wettability and suppressing interface recombination. Furthermore, compared to the small-molecular DBPP, Poly-DBPP exhibits higher conductance and excellent uniformity. This translates to a remarkable power conversion efficiency of 25.1% for blade-coated PSCs and 22.0% for large-area modules, respectively. Additionally, the PSCs based on Poly-DBPP demonstrate impressive operational stability, retaining 92% of their initial PCE after 1,600 h of light soaking. This work presents a promising strategy for designing multifunctional hole transporters, paving the way for highly efficient and stable PSCs.

Exploring the Potential and Hurdles of Perovskite Solar Cells with p-i-n Structure.

The p-i-n architecture within perovskite solar cells (PSCs) is swiftly transitioning from an alternative concept to the forefront of perovskite photovoltaic technology, driven by significant advancements in performance and suitability for tandem solar cell integration. The relentless pursuit to increase efficiencies and understand the factors contributing to instability has yielded notable strategies for enhancing p-i-n PSC performance. Chief among these is the advancement in passivation techniques, including the application of self-assembled monolayers (SAMs), which have proven central to mitigating interface-related inefficiencies. This Perspective delves into a curated selection of recent impactful studies on p-i-n PSCs, focusing on the latest material developments, device architecture refinements, and performance optimization tactics. We particularly emphasize the strides made in passivation and interfacial engineering. Furthermore, we explore the strides and potential of p-i-n structured perovskite tandem solar cells. The Perspective culminates in a discussion of the persistent challenges facing p-i-n PSCs, such as long-term stability, scalability, and the pursuit of environmentally benign solutions, setting the stage for future research directives.

Conjugated Group Tuning of Self‐Assembled Monolayer for Efficient Hole‐Transport Layer in Organic Solar Cells

AbstractP‐type carbazole‐derived self‐assembled monolayers (SAMs) have garnered significant attention as promising hole transport layers (HTLs) in the development of highly efficient organic solar cells (OSCs). However, it still lacks the effective navigation to modulate the terminal functional groups of SAMs to achieve a compromise between the highest occupied molecular orbital (HOMO) energy levels and self‐aggregation behavior. Herein, the terminal functional groups are adjusted and three SAMs are synthesized, namely, t‐Bu‐3PACz, Ph‐3PACz, and Bz‐3PACz to comprehensively investigate their intrinsic properties and influence on photovoltaic performance. Among them, Ph‐3PACz featuring an exceptionally suitable conjugated region and steric hindrance exhibits the best compatibility with the active layer, superior electrical conductivity, HOMO energy level aligning with polymer donor, and ordered film packing. As a result, the photovoltaic devices based on Ph‐3PACz exhibit an open‐circuit voltage (VOC) of 0.850 V, a short‐circuit current density (JSC) of 28.7 mA cm−2, and a fill factor (FF) of 78.5%, thus resulting in a remarkable power conversion efficiency (PCE) of 19.2%. This work provides an effective and easily navigable method to modulate the molecular packing and energy levels of SAMs, thereby achieving highly efficient OSCs.

Surface Functionalized Plasmonic Sensors for Uric Acid Detection With Gold-Graphene Stacked Nanocomposites

Abstract This study presented a surface-functionalized sensor probe using 3-aminopropyltriethoxysilane (APTES) self-assembled monolayers on a Kretschmann-configured plasmonic platform. The probe featured stacked nanocomposites of gold (via sputtering) and graphene quantum dots (GQD, via spin-coating) for highly sensitive and accurate uric acid (UA) detection within the physiological ranges. Characterization encompassed the field emission scanning electron microscopy for detailed imaging, energy-dispersive X-ray spectroscopy for elemental analysis, and Fourier transform infrared spectroscopy for molecular identification. Surface functionalization increased sensor sensitivity by 60.64%, achieving 0.0221 °/(mg/dL) for the gold-GQD probe and 0.035 5 °/(mg/dL) for the gold-APTES-GQD probe, with linear correlation coefficients of 0.8249 and 0.8509, respectively. The highest sensitivity was 0.070 6 °/(mg/dL), with a linear correlation coefficient of 0.993 and a low limit of detection of 0.2 mg/dL. Furthermore, binding affinity increased dramatically, with the Langmuir constants of 14.29 µM−1 for the gold-GQD probe and 0.000 1 µM−1 for the gold-APTES-GQD probe, representing a 142 900-fold increase. The probe demonstrated notable reproducibility and repeatability with relative standard deviations of 0.166% and 0.013%, respectively, and exceptional temporal stability of 99.66%. These findings represented a transformative leap in plasmonic UA sensors, characterized by enhanced precision, reliability, sensitivity, and increased surface binding capacity, synergistically fostering unprecedented practicality.

Abstract 4134630: Focal Delivery of Antiplatelet Therapy to Prevent Post-Stent Bleeding in High-Risk Patients

Background: Over 3 million percutaneous coronary interventions (PCI) are performed yearly to treat coronary artery stenosis. Stent thrombosis is a catastrophic complication associated with high morbidity and mortality, and its prevention requires the use of prolonged dual antiplatelet therapy (DAPT), which increases bleeding risk. Within the first 30 days after PCI, the mortality of DAPT-associated major bleeding is on par with recurrent myocardial infarction. Research question: No stent system that provides focal antiplatelet activity to prevent stent thrombosis exists, eliminating the need for systemic DAPT and subsequent bleeding risk. Aim: To address this unmet need, we have developed such a stent, termed “the ticagrelor coated stent” (TCS). Methods: Self-assembled monolayers (SAMs) of 12-aminododecylphosphonic acid (ADPA) were formed on cobalt-chromium stents. The amine tail group of ADPA was used to link the ticagrelor molecule through a Mitsunobu reaction and confirmed via infrared spectroscopy. Coating uniformity was validated via atomic force microscopy. In an ex-vivo porcine arterio-venous fistula model, the ticagrelor-coated stents (TCS) were placed in alternating series adjacent to uncoated bare metal stents (BMS). Similarly, TCS and everolimus-eluting stents were placed in the porcine left circumflex arteries for acute (7 days) and chronic (35 days) studies. Results: ( Figure 1 ): Grossly, no thrombus was seen on the TCS compared to the BMS. Platelet and micro-thrombi adherence were significantly reduced on TCS. Notably, inflammation, measured by neutrophil and monocyte adherence, was also reduced by approximately 10-fold on the TCS vs the BMS. Angiography, optical coherence tomography (OCT), and histopathology results show the TCS widely patent without systemic DAPT. Conclusion: These findings show that TCS prevents stent thrombosis through focal anti-platelet action and may reduce the bleeding risk associated with prolonged use of systemic DAPT. Long-term safety and efficacy studies are underway.

Engineering DNA Nanocube SAM Scaffolds for FRET-Based Biosensing: Interfacial Characterization and Sensor Demonstration.

Decorating a gold surface with molecular-level control over the positioning of DNA probes was demonstrated using a self-assembled monolayer (SAM) of wireframe DNA nanocube structures. The DNA nanocubes were specifically adsorbed and oriented using thiol-modified DNA on one face of the cube. The DNA nanocube SAM had a uniform coverage over the gold single crystal bead electrode with a separation of 20-30 nm measured by AFM. The face of the nanocube furthest from the gold surface was designed to hybridize with two different sequences of a 50 base single-stranded DNA probe that was modified with a fluorophore. The first 20 bases were hybridized with the DNA nanocube. One of a pair of FRET fluorophores was used for each probe strand. The dimensions of the nanocube controlled the relative spacing between these fluorophores. When the DNA probes were single-stranded, a FRET signal was observed. FRET decreased to background levels when a complementary DNA target was hybridized to either probe, resulting in a turn-off sensor with little cross-talk between the individual hybridization events. Hybridization isotherms for one target gave KA = 170 pM and a detection limit <50 pM. In addition, the DNA nanocube SAM was configured to be used as a turn-on NeutrAvidin sensor using biotinylated DNA targets hybridized to each probe resulting in an increase in FRET. We show that the wireframe DNA nanocube can be an effective scaffold for preparing biosensors with controlled separation between surface-bound probes facilitating precise sensor surface design and enabling a wide range of sensing modalities with more than one signal available for correlative confirmation of the target binding.

Thermoelectric Properties of Benzothieno-Benzothiophene Self-Assembled Monolayers in Molecular Junctions.

We report a combined experimental (C-AFM and SThM) and theoretical (DFT) study of the thermoelectric properties of molecular junctions made of self-assembled monolayers on Au of thiolated benzothieno-benzothiophene (BTBT) and alkylated BTBT derivatives (C8-BTBT-C8). We measure the thermal conductance per molecule at 15 and 8.8 pW/K, respectively, among the lowest values for molecular junctions so far reported (10-50 pW/K). The lower thermal conductance for C8-BTBT-C8 is consistent with two interfacial thermal resistances introduced by the alkyl chains, which reduce the phononic thermal transport in the molecular junction. The Seebeck coefficients are 36 and 245 μV/K, respectively, the latter due to the weak coupling of the core BTBT with the electrodes. We deduce a thermoelectric figure of merit ZT up to ≈10-4 for the BTBT molecular junctions at 300 K, on a par with the values reported for archetype molecular junctions (oligo(phenylene ethynylene) derivatives).

Mitigating coherent loss in superconducting circuits using molecular self-assembled monolayers

In planar superconducting circuits, decoherence due to materials imperfections, especially two-level-system (TLS) defects at different interfaces, is a primary hurdle for advancing quantum computing and sensing applications. Traditional methods for mitigating TLS loss, such as etching oxide layers at metal and substrate interfaces, have proven to be inadequate due to the persistent challenge of oxide regrowth. In this work, we introduce a novel approach that employs molecular self-assembled monolayers (SAMs) to chemically bind at different interfaces of superconducting circuits. This technique is specifically tested here on coplanar waveguide (CPW) resonators, in which this method not only impedes oxide regrowth after surface etching but can also tailors the dielectric properties at different resonators interfaces. The deployment of SAMs results in a consistent improvement in the measured quality factors across multiple resonators, surpassing those with only oxide-etched resonators. The efficiency of our approach i3s supported by microwave measurements of multiple devices conducted at millikelvin temperatures and correlated with detailed X-ray photoelectron spectroscopy (XPS) and transmission electron microscopy (TEM) characterizations of SAM-passivated resonators. The compatibility of SAMs materials with the established fabrication techniques offers a promising route to improve the performance of superconducting quantum devices.

Stereo-Hindrance Induced Conformal Self-Assembled Monolayer for High Efficiency Inverted Perovskite Solar Cells.

Self-assembled monolayers (SAMs) are employed as hole-selective contacts in inverted perovskite solar cells (PSCs) and have achieved record power conversion efficiency (PCE) over 26%. However, the tendency of extensively employed SAM [2-(3,6-Dimethoxy-9H-carbazol-9-yl)ethyl]phosphonic acid to aggregate leads to its uneven coverage to the transparent conducting oxide substrate, which subsequently compromises the photovoltaic performance. Herein, a novel tert-butyl functionalized phosphonic acid carbazole SAM is developed, i.e., (4-(3,6-di-tert-butyl-9H-carbazol-9-yl)butyl)phosphonic acid (tBu-4PACz), and introduced to a mixed SAM system as the hole-extraction layer in inverted PSCs. The stereo-hindrance of the bulky tert-butyl group prevents undesired aggregation and leads to better conformality, which facilitates more efficient hole-extraction and suppresses interfacial recombination losses. The tBu-4PACz SAM-based inverted PSC has achieved record level PCE of 26.25% (26.21%, certificated) with outstanding fill factors over 86%. Moreover, the mixed SAM based inverted PSC devices maintained over 94.7% of their initial efficiency after 500h continuous maximum power-point tracking under simulation 1-sun irradiation.

Ethanol Processable Inorganic‐Organic Hybrid Hole Transporting Layers Enabled 20.12 % Efficiency Organic Solar Cells

AbstractIn this study, a high‐performance inorganic‐organic hybrid hole transporting layer (HTL) was developed using ethanol‐soluble alkoxide precursors and a self‐assembled monolayer (SAM). Three metal oxides‐vanadium oxide (VOx), niobium oxide (Nb2O5), and tantalum oxide (Ta2O5)‐were synthesized through successive low‐temperature (100 °C) thermal annealing (TA) and UV‐ozone (UVO) treatments of their respective precursors: vanadium oxytriethoxide (EtO−V), niobium ethoxide (EtO−Nb), and tantalum ethoxide (EtO−Ta). Among these, the Nb2O5 film exhibited excellent transmittance, a high work function, and good conductivity, along with a more compact and uniform structure featuring fewer interfacial defects, which facilitated efficient charge extraction and transport. Furthermore, the deposition of a SAM of (2‐(9H‐carbazol‐9‐yl)ethyl)phosphonic acid (2PACz) on top of Nb2O5 further passivated defects, enhancing interfacial contact with the photoactive layer. The resulting inorganic‐organic hybrid HTL of Nb2O5/2PACz demonstrated excellent compatibility with various photoactive blends, achieving impressive power conversion efficiencies of 19.44 %, 19.18 %, and 20.12 % for the PM6:L8‐BO, PM6:BTP‐eC9, and D18:BTP‐eC9 based organic solar cells, respectively. 20.12 % is the best performance for bulk heterojunction organic solar cells with binary components as the photoactive layer.

Investigating Molecular Junctions Based on Mixed Self-Assembled Monolayers to Understand the Impact of Intermolecular Interactions on Transport.

To interrogate the importance of intermolecular interactions on charge transport at the nanoscale, we investigate molecular tunnel junctions based on mixed self-assembled monolayers (SAMs) of 1-alkyl (CnT) thiols and their fluorinated counterparts (F-CnT) that have substantially different tunneling conductances. Experiments on mixed CnT1-x:F-CnTx SAMs between Au contacts reveal a strongly nonlinear (exponential) dependence of the tunneling conductance G on composition x, a behavior that is tempting to assign to the strong impact of intra-SAM intermolecular interactions. However, analysis suggests that the exponential dependence of G on x does not arise from intra-SAM intermolecular interactions, but instead emerges from the work function modification of the Au electrode which varies linearly with x.

Barrier height modulation in Amorphous IGZO/Metal interface using Trichlorosilane-Based Self-Assembled monolayers

Amorphous InGaZnO (a-IGZO) has become a key material for thin-film transistors (TFTs) due to its high mobility, low leakage current, and optical transparency. However, the increasing demands for ultra-high-resolution displays, such as those required for virtual and augmented reality applications, pose challenges related to contact resistance and Fermi level pinning at the a-IGZO/metal interface, as TFTs are scaled down. To address these challenges, we investigated trichlorosilane-based self-assembled monolayers (SAMs) as an interfacial layer between a-IGZO and metal contacts (Al, Ag, and Ni). By employing SAMs with varying carbon chain lengths—methyl trichlorosilane (MTS), octyl trichlorosilane (OTS), and octadecyl trichlorosilane (ODTS)—we aimed to improve interface properties and reduce contact resistance. Our study comprehensively analyzed thermionic emission characteristics, including barrier height, saturation current, ideality factor, and contact resistance, across different metal electrodes. The results show that the SAM layers effectively modulate barrier height and Fermi level pinning, depending on the combination of SAM layers (MTS, OTS, and ODTS) and metal electrodes (Al, Ag, and Ni). Additionally, the study provides insights into the influence of SAM carbon chain length on direct tunneling current and interface passivation, and contributes to a deeper understanding of conduction mechanisms at a-IGZO/metal interfaces.

Homogeneous crystallization of MA-free, wide-bandgap perovskite films via self-assembled monolayer capping for laminated silicon/perovskite tandem solar cells

Methylammonium (MA)-free wide-bandgap perovskite films with improved thermal/light stabilities have gained considerable attention for its application in silicon/perovskite tandem solar cells. However, compared to their MA counterparts, MA-free films often suffer from fast crystallization, resulting in uneven surface morphology, poor crystalline features, and excess PbI2 impurities. To modulate the crystallization dynamics of MA-free wide-bandgap Cs0.22FA0.78Pb(Br0.15I0.85)3 films, this study proposes a self-assembled monolayer (SAM)-capped crystallization strategy, wherein C42H52N6O4RuS2 (Z907) is added to the isopropanol antisolvent in the one-step solution process. The optimized Cs0.22FA0.78Pb(Br0.15I0.85)3 film has a wide bandgap of 1.68 eV with superior morphological/crystalline properties, enhanced thermal/moisture stability, and improved interfacial energy‐level alignment. These favorable characteristics are attributed to the efficient coupling between the positive ion vacancies in the precursor, the −SCN and –COOH functional groups in Z907, and the superhydrophobic nonyl chains in the Z907 SAM. The semitransparent perovskite solar cells (PSCs) built with the optimized Cs0.22FA0.78Pb(Br0.15I0.85)3 films demonstrate substantially high power conversion efficiencies (PCEs) of 21.63 %@0.09 cm2 and 19.12 %@1 cm2. Furthermore, based on the high-efficiency PSCs, the corresponding two-terminal laminated silicon/perovskite tandem solar cells achieved record-high PCEs of 32.07 %@0.09 cm2 and 28.32 %@1 cm2 with negligible hysteresis and improved humidity/thermal stability.

Dimensional analysis of diffusive association rate equations

Diffusive adsorption/association is a fundamental step in almost all chemical reactions in diluted solutions, such as organic synthesis, polymerization, self-assembly, biomolecular interactions, electrode dynamics, catalysis, chromatography, air and water environmental dynamics, and social and market dynamics. However, predicting the rate of such a reaction is challenging using the equations established over 100 years ago. Several orders of magnitude differences between the theoretical predictions and experimental measurements for various systems, from self-assembled monolayers to protein-protein aggregations, make such calculations meaningless in many situations. I believe the major problem is that the time-dependent evolution curve of Fick’s gradient is an ideal assumption in most cases, and its slope is significantly overestimated. This paper digs into Fick’s gradient problem for 3D cases and provides a solution using the single-molecule diffusion probability density function discretely.