Monodisperse Emulsions Research Articles

Precision Macroporous Monoliths Made Using High‐Throughput Microfluidic Emulsification

Materials with pore sizes ranging from micrometers to millimeters find use in thermal insulation, acoustics, separation, and energy‐related applications. While several routes have been developed to manufacture such macroporous materials, current fabrication technologies remain limited in either scalability or pore size control. Herein, a scalable microfluidic approach is reported to create macroporous materials with precisely controlled pore sizes from monodisperse emulsion templates. Emulsion droplets produced in a parallelized step emulsification device are assembled by gravity and converted into macroporous monoliths by polymerization and drying. Microstructural analysis and model filtration experiments show that the pore windows of the bulk macroporous material can be tightly controlled and used for the size‐selective separation of particles from suspensions. By heat treating macroporous structures of selected compositions, one can also create bulk carbon and silica glass monoliths with unique cellular architectures for potential applications as filters, separation membranes, or catalyst supports.

Upscaled Production of Satellite-Free Droplets: Step Emulsification with Deterministic Lateral Displacement.

Step emulsification is a key technique for achieving scalable production of monodisperse emulsion droplets owing to its resilience to flow fluctuations. However, the persistent issue of satellite droplets, an inherent byproduct of main droplets, poses challenges for achieving truly uniform product sizes. In a previous study, we introduced a module with step-emulsifier nozzles upstream and deterministic lateral displacement (DLD) micropillar arrays downstream to generate satellite-free droplets at a low throughput. In this study, we demonstrate an upscaled parallelized setup with ten modules that were designed to produce satellite-free droplets. Each module integrated 100 step-emulsification nozzles in the upstream region with DLD micropillar arrays downstream. We conducted 3D flow simulations to ensure homogeneous distribution of the input fluids. Uniformly supplying an aqueous polyvinyl alcohol solution and an acrylate monomer as continuous and dispersed phases into the ten modules, the nozzles in each module exhibited a production rate of 539.5 ± 28.6 drop/s (n = 10). We successfully isolated the main droplets with a mean diameter of 66 μm and a coefficient of variation of 3.1% from satellite droplets with a mean diameter of 3 μm. The total throughput was 3.0 mL/h. The high yield and contamination-free features of our approach are promising for diverse industrial applications.

Size prediction of drug-loaded Polymeric (PLGA) microparticles prepared by microfluidics

Drug-loaded poly (lactic-co-glycolic acid) (PLGA) microparticles with ex-act control over size and homogeneity were created using a glass microfluidic cross-junction droplet generator. A numerical simulation based on the volume-of-fluid (VOF) method is used to study the behavior of two immiscible fluids that are injected into the microdevice: hydrophobic drug and (PLGA) in dichloromethane (DCM) serving as the dispersed phase and aqueous solution of poly (vinyl alcohol) (PVA) as the continuous phase. The simulation predicts the created emulsion droplet size regulation by adjusting the phase's flow rate. In a procedure known as droplet shrinking, dichloromethane extraction, and evaporation took place to further transform the monodisperse emulsion into PLGA microparticles. A numerical relationship is also given for the variations in the volume of droplets during shrinkage. The numerical results were consistent with the experiments. To further illustrate this technique's ability to manufacture designed microspheres with the least amount of repeated experiments, as well as the importance of the droplet microfluidic approach in terms of regulating diameter and monodispersity, opioid antagonist naltrexone is loaded in PLGA microspheres. According to the findings, integrating a versatile microfluidic approach with numerical methodologies enables the development of more reliable and reproducible drug delivery systems.

Tailored surface wettability and pore structure of hydrophobic SiO2/SiC membranes for preparing monodisperse emulsion with high-efficiency

The combustion efficiency of diesel can be significantly improved in the form of emulsion, contributing to the reduced emission of nitrogen oxides. It is thus worthwhile to develop an energy-efficient way to prepare stable diesel emulsions with uniform droplet size. Ceramic membranes with hydrophobic surface are highly desirable for high-efficient membrane emulsification process. We proposed a novel and straightforward approach of immersing silicon carbide (SiC) membrane into hydrophobic silica solutions obtained by co-hydrolysis to tailoring hydrophobic SiO2/SiC membranes. The effects of methyltriethoxysilane addition and aging days on the hydrophobicity of these silica particles were systematically investigated. Then, the deposition of silica particles on the SiC substrates was further controlled by regulating the dipping duration and dipping times. The membranes (M2-SiO2/SiC) prepared with 240 s and dipping 4 times showed an increased contact angle (136°), mean pore size of 0.4 μm, open porosity of 30 %, and pure water permeance of ∼ 1500 L·m−2·h−1·bar−1. Moreover, the modified membranes were demonstrated to be mechanically stable. Furthermore, monodisperse diesel emulsions with an average droplet size of 2.1 μm and droplet size distribution of 0.61 can be generated by using the optimized M2-SiO2/SiC membranes at high emulsifying flux (1910 L·m−2·h−1). This work proposed an efficient strategy to simultaneously optimize the surface wettability and pore structure of ceramic membranes, satisfactorily for membrane emulsification applications with high efficiency.

Droplet size dependency and spatial heterogeneity of lipid oxidation in whey protein isolate-stabilized emulsions

Spatiotemporal assessment of lipid and protein oxidation is key for understanding quality deterioration in emulsified food products containing polyunsaturated fatty acids. In this work, we first mechanistically validated the use of the lipid oxidation-sensitive fluorophore BODIPY 665/676 as a semi-quantitative marker for local peroxyl radical formation. Next, we assessed the impact of microfluidic and colloid mill emulsification (respectively producing mono- and polydisperse droplets) on local protein and lipid oxidation kinetics in whey protein isolate (WPI)-stabilized emulsions. We further used BODIPY 581/591 C11 and CAMPO-AFDye 647 as colocalisation markers for lipid and protein oxidation. The polydisperse emulsions showed an inverse relation between droplet size and lipid oxidation rate. Further, we observed less protein and lipid oxidation occurring in similar sized droplets in monodisperse emulsions. This observation was linked to more heterogeneous protein packing at the droplet surface during colloid mill emulsification, resulting in larger inter-droplet heterogeneity in both protein and lipid oxidation. Our findings indicate the critical roles of emulsification methods and droplet sizes in understanding and managing lipid oxidation.

Unravelling the effect of droplet size on lipid oxidation in O/W emulsions by using microfluidics

Lipid oxidation in emulsions is hypothesised to increase with decreasing droplet size, as this increases the specific oil–water interfacial area, where lipid oxidation is expected to be initiated. In literature, however, contradictory results have been reported, which can be caused by confounding factors such as the oil droplet polydispersity and the distribution of components between the available phases. In this work, monodisperse surfactant-stabilised emulsions with highly controlled droplet sizes of 4.7, 9.1, and 26 µm were produced by microfluidic emulsification. We show that lipid oxidation increases with decreasing droplet size, which we ascribe to the increased contact area between lipids and continuous phase prooxidants. Besides, a significant amount of oxygen was consumed by oxidation of the surfactant itself (Tween 20), an effect that also increased with decreasing droplet size. These insights substantiate the importance of controlling droplet size for improving the oxidative stability of emulsions.

A Microchannel Device for Droplet Classification by Manipulation Using Piezoelectric Vibrator

Emulsion formulations should be monodispersed in terms of their stability. Therefore, there is a need for a device that can classify droplets of the desired size from polydispersed emulsions in a fluidized bed manufacturing system. In the previous study, we evaluated the fabrication of a droplet manipulation device using acoustic radiation forces through simulation using the finite element method. In this study, particle manipulation experiments using 1, 6, and 10 µm polystyrene particles were first estimated and evaluated in comparison with their theoretical particle behavior. Based on the results we obtained, the driving conditions and droplet behavior were derived, and the droplet manipulation device using ultrasonic waves to shrink monodisperse emulsions was evaluated. As a result, the droplet classification effect in the microchannel was confirmed to be consistent with the droplet behavior prediction, and the microchannel structure with a constriction component improved its classification effect.

Capillary imbibition of confined monodisperse emulsions in microfluidic channels.

We study imbibition of a monodisperse emulsion into high-aspect ratio microfluidic channels with the height h comparable to the droplet diameter d. Two distinct regimes are identified in the imbibition dynamics. In a strongly confined system (the confinement ratio d/h = 1.2 in our experiments), the droplets are flattened between the channel walls and move more slowly compared to the average suspension velocity. As a result, a droplet-free region forms behind the meniscus (separated from the suspension region by a sharp concentration front) and the suspension exhibits strong droplet-density and velocity fluctuations. In a weaker confinement, d/h = 0.65, approximately spherical droplets move faster than the average suspension flow, causing development of a dynamically unstable high-concentration region near the meniscus. This instability results in the formation of dense droplet clusters, which migrate downstream relative to the average suspension flow, thus affecting the entire suspension dynamics. We explain the observed phenomena using linear transport equations for the particle-phase and suspension fluxes driven by the local pressure gradient. We also use a dipolar particle interaction model to numerically simulate the imbibition dynamics. The observed large velocity fluctuations in strongly confined systems are elucidated in terms of migration of self-assembled particle chains with highly anisotropic mobility.

Emergence of preferential flow paths and intermittent dynamics in emulsion transport in porous media.

We investigate the dynamics of emulsions within a two-dimensional porous medium using an integrated experimental approach that combines pore-level dynamics of single emulsions and bulk transport properties of the medium. Using an on-chip microfluidic drop-maker, we precisely control the concentration and sizes of emulsions injected into the medium. The dynamics of emulsion droplets are highly intermittent despite a small average velocity over the trajectory of an individual emulsion. At low concentrations, emulsions predominantly flow through pores with higher local velocities including pores smaller than the size of emulsion droplets, leading to trapping of emulsions and a decrease in medium porosity. Preferential pathways for the emulsions emerge within the medium once the porosity of the medium decreases significantly, from 55% to 36%. At constant injection flow rates and low concentrations of monodisperse emulsions, these pathways remain the only paths of transport of emulsions within the medium. Introducing a slight polydispersity in emulsion sizes unveiled additional transport pathways. Our pore-level measurements reveal that the average velocity of emulsions scales with the inverse residence time of an emulsion, and this scaling separates the emulsions into distinct groups along the emergent preferential pathways.

Linking processing to formulation in dynamic membranes of tunable pore size for lemon oil emulsions

Lemon oil is widely used for food flavoring purposes, but due to its low water solubility and tendency to oxidation, requires encapsulation in oil-in-water (O/W) emulsions. Therefore, this study assesses the use of two natural biopolymers and a tailor-made one to encapsulate lemon oil with a dynamic membrane of tunable pore size (DMTS). This emulsification system, consisting of an outer bed layer of silica beads sitting upon a metal microporous metallic membrane, overcomes the limitations of conventional emulsification membranes, such as low throughput, complex cleaning before reuse, fixed membrane thickness and pore size, and significative protein fouling. The effects of the DMTS characteristics on the droplet size distribution and emulsion productivity were studied. The results demonstrate that mono-dispersed emulsions can be obtained using dynamic membranes for all the biopolymers studied and the productivity can reach 1000 m3 m−2h−1 pointing out the suitability of this system for scale-up.

Soft glassy flow of highly concentrated monodisperse emulsions in microchannels: Layered structure and stairwise velocity profile

Concentrated emulsions with a jammed structure exhibit intricate flow properties, which are not fully comprehended. The flow behavior of a highly concentrated monodisperse emulsion in a microchannel is investigated using dissipative particle dynamics simulations. The local flow curves are analyzed at different sampling resolutions and tracer sizes. The characteristics of the velocity profile obtained at low resolutions agree with experimental observations. However, at high resolutions, the velocity profile deviates from the Herschel-Bulkley model, and stairwise changes appear in the shear zone. Droplet distributions indicate that the layered structure forms within the shear zone and expands with increasing pressure drop. Due to the infrequent exchange of droplets between neighboring layers, a sudden change in velocity arises between them. Additionally, at high pressure drops, the previously uniform droplet distribution in the plug zone transforms into a layered structure as well. Our findings suggest that highly concentrated emulsions under confinement exhibit non-homogeneous structures which affect their flow behavior significantly.

Recent Developments in the Viscosity Modeling of Concentrated Monodisperse Emulsions.

Emulsions form a large group of food materials. Many foods are either partly or wholly emulsions or are in the form of emulsion at some stage of the production process. A good understanding of the rheological properties of emulsions, especially their shear viscosity, is essential in the design, formulation, and processing of food emulsions. The texture and mouthfeel of food emulsions are also largely influenced by emulsion viscosity. Therefore, it is of practical importance to be able to correlate and predict emulsion viscosity as a function of droplet concentration and other relevant variables. In this article, the recent developments made in the viscosity modeling of concentrated emulsions are reviewed. The viscosity models for concentrated emulsions published in the twenty-first century are discussed, compared, and evaluated using a large body of experimental viscosity data available on emulsions. The effects of droplet size distribution and capillary number on the viscosity of concentrated emulsions are also discussed in detail. A new generalized viscosity model is developed for concentrated emulsions that includes the effect of capillary number and is accurate with small average percent relative error (within 3%).

Ultrasound treated fish myofibrillar protein: Physicochemical properties and its stabilizing effect on shrimp oil-in-water emulsion

Effects of ultrasonication at different amplitudes (40% and 60%) and time (5, 10, and 15 min) on the physicochemical and emulsifying properties of the fish myofibrillar protein (FMP) were investigated. Solubility, surface hydrophobicity, and emulsifying properties were augmented when FMP was subjected to ultrasonication at 40% amplitude for 15 min (p < 0.05). Protein pattern study revealed that augmenting amplitude and duration of ultrasound treatment reduced band intensity of myosin heavy chain. Ultrasound treatment facilitated the adsorption of FMP on oil droplets as indicated by the increases in both adsorbed and interfacial protein contents (p < 0.05). Ultrasound-treated FMP (UFMP) sample showed the alteration in chemical bonds as depicted by Fourier transform infrared (FTIR) spectra. Ultrasound treatment altered the β-sheet and random coil of FMP. During storage for 30 days at 30 °C, UFMP stabilized shrimp oil (SO)-in-water emulsion had higher turbidity but lower d32, d43, and polydispersity index than emulsion stabilized by untreated FMP (p < 0.05). Furthermore, emulsion stabilized by UFMP had lower flocculation and coalescence indices (p < 0.05). Microstructure observation revealed smaller droplet sizes and higher stability of droplets in emulsion stabilized by UFMP. Confocal laser scanning microscopic images demonstrated a monodisperse emulsion stabilized by UFMP. This coincided with higher viscosity and modulus values (G' and G″ ). Emulsion stabilized by UFMP exhibited viscous, shear-thinning, and non-Newtonian behavior and no phase separation occurred during storage. Therefore, ultrasonication was proven to be a potential method for enhancing the emulsifying properties of FMP and improving the stability of SO-in-water emulsion during prolonged storage.

Fabrication of monodisperse droplets and microcapsules using microfluidic chips: a review of methodologies and applications

Abstract Microfluidics has been applied in the preparation of monodisperse droplets and microcapsules due to its high encapsulation efficiency, its ability to create uniform particle sizes, and its capacity to control core–shell ratio and structure. To bring to the fore methodologies for the fabrication and application of monodisperse microcapsules using microfluidics, we present a review of the design, structure, materials, and surface modification techniques of various microfluidic chips. The review also covers fabrication methods, operating parameters and regulation methods of single and multiple monodisperse emulsion droplets fabricated from various microfluidic devices. Our findings show that particle size of monodisperse droplets depend mainly on microchannel characteristic size and flow rate, with particle size increasing with larger microchannel but decreasing with higher continuous phase flow rate. We additionally reviewed and compared various fabrication methods for monodisperse microcapsules, such as interfacial polymerization, free-radical polymerization, ionic cross-linking, and solvent evaporation. We further reviewed and examined the application of monodisperse microcapsules in biology applications, food engineering, composite materials development, and pharmaceutical industry. We found that high-throughput microfluidics for scale-up monodisperse microcapsule preparation towards uniform degradation and targeted release properties of monodisperse microcapsules would be key innovative direction for future applications.

Nonequilibrium structure formation in electrohydrodynamic emulsions.

Application of an electric field across the interface of two fluids with low, but non-zero, conductivities gives rise to a sustained electrohydrodynamic (EHD) fluid flow. In the presence of neighboring drops, drops interact via the EHD flows of their neighbors, as well as through a dielectrophoretic (DEP) force, a consequence of drops encountering disturbance electric fields around their neighbors. We explore the collective dynamics of emulsions with drops undergoing EHD and DEP interactions. The interplay between EHD and DEP results in a rich set of emergent behaviors. We simulate the collective behavior of large numbers of drops; in two dimensions, where drops are confined to a plane; and three dimensions. In monodisperse emulsions, drops in two dimensions cluster or crystallize depending on the relative strengths of EHD and DEP, and form spaced clusters when EHD and DEP balance. In three dimensions, chain formation observed under DEP alone is suppressed by EHD, and lost entirely when EHD dominates. When a second population of drops are introduced, such that the electrical conductivity, permittivity, or viscosity are different from the first population of drops, the interaction between the drops becomes non-reciprocal, an apparent violation of Newton's Third Law. The breadth of consequences due to these non-reciprocal interactions are vast: we show selected cases in two dimensions, where drops cluster into active dimers, trimers, and larger clusters that continue to translate and rotate over long timescales; and three dimensions, where drops form stratified chains, or combine into a single dynamic sheet.

Preparation of monodisperse water-in-oil emulsions using microchannel homogenization

The droplet size and uniformity of water-in-oil (W/O) emulsions are important properties governing their stability in diverse applications. Monodisperse emulsions are preferred over polydisperse emulsions because their droplet size and distribution are easier to control. Here, we prepared (relatively) monodisperse W/O emulsions using microchannel homogenization (MCH) with an asymmetric straight-through microchannel (MC) array chip. Numerous asymmetric straight-through MCs, each comprising a microslot and circular microhole (diameter: 10 µm), were microfabricated on a silicon-on-insulator chip. We investigated the effects of emulsifier concentration, continuous-phase viscosity, dispersed-phase volume fraction, and the operating pressure of coarse W/O emulsions on the preparation of W/O emulsions, droplet size, and size distribution. The continuous phase was composed of soybean or medium-chain triglyceride oil solutions with tetraglycerin monolaurate condensed ricinoleic acid ester as the emulsifier. The dispersed phase was a Milli-Q water solution containing NaCl and/or polyethylene glycol (m.w. 20,000). The results demonstrate the suitability of MCH for the preparation of monodisperse W/O emulsions with an average droplet diameter of 12–15 µm and 8% lowest coefficient of variation using hydrophobically surface-treated asymmetric straight-through MCs with appropriately selected emulsion compositions and operating conditions.

Inferring the stability of concentrated emulsions from droplet configuration information

When droplets are tightly packed in a 2D microchannel, coalescence of a pair of droplets can trigger an avalanche of coalescence events that propagate through the entire emulsion. This propagation is found to be stochastic, i.e. every coalescence event does not necessarily trigger another. To study how the local probabilistic propagation affects the dynamics of the avalanche, as a whole, a stochastic agent based model is used. Taking as input, i) how the droplets are packed (configuration) and ii) a measure of local probabilistic propagation (experimentally derived; function of fluid and other system parameters), the model predicts the expected size distribution of avalanches. In this article, we investigate how droplet configuration affects the avalanche dynamics. We find the mean size of these avalanches to depend non-trivially on how droplets are packed together. Large variations in the avalanche dynamics are observed when droplet packing are different, even when the other system properties (number of droplets, fluid properties, channel geometry, etc.) are kept constant. Bidisperse emulsions show less variation in the dynamics and they are surprisingly more stable than monodisperse emulsions. To get a systems-level understanding of how a given droplet-configuration either facilitates or impedes the propagation of an avalanche, we employ a graph-theoretic analysis, where emulsions are expressed as graphs. We find that the properties of the underlying graph, namely the mean degree and the algebraic connectivity, are well correlated with the observed avalanche dynamics. We exploit this dependence to derive a data-based model that predicts the expected avalanche sizes from the properties of the graph.

Microfluidic Particle Engineering of Hydrophobic Drug with Eudragit E100─Bridging the Amorphous and Crystalline Gap.

Co-processing active pharmaceutical ingredients (APIs) with excipients is a promising particle engineering technique to improve the API physical properties, which can lead to more robust downstream drug product manufacturing and improved drug product attributes. Excipients provide control over critical API attributes like particle size and solid-state outcomes. Eudragit E100 is a widely used polymeric excipient to modulate drug release. Being cationic, it is primarily employed as a precipitation inhibitor to stabilize amorphous solid dispersions. In this work, we demonstrate how co-processing of E100 with naproxen (NPX) (a model hydrophobic API) into monodisperse emulsions via droplet microfluidics followed by solidification via solvent evaporation allows the facile fabrication of compact, monodisperse, and spherical particles with an expanded range of solid-state outcomes spanning from amorphous to crystalline forms. Low E100 concentrations (≤26% w/w) yield crystalline microparticles with a stable NPX polymorph distributed uniformly across the matrix at a high drug loading (∼89% w/w). Structurally, E100 incorporation reduces the size of primary particles comprising the co-processed microparticles in comparison to neat API microparticles made using the same technique and the as-received API powder. This reduction in primary particle size translates into an increased internal porosity of the co-processed microparticles, with specific surface area and pore volume ∼9 times higher than the neat API microparticles. These E100-enabled structural modifications result in faster drug release in acidic media compared to neat API microparticles. Additionally, E100-NPX microparticles have a significantly improved flowability compared to neat API microparticles and as-received API powder. Overall, this study demonstrates a facile microfluidics-based co-processing method that broadly expands the range of solid-state outcomes obtainable with E100 as an excipient, with multiscale control over the key attributes and performance of hydrophobic API-laden microparticles.

Changes in Emulsifying and Physical Properties of Shrimp Oil/Soybean Oil‐in‐Water Emulsion Stabilized by Fish Myofibrillar Protein during the Storage

AbstractProperties of shrimp oil/soybean oil (30:70)‐in‐water emulsion stabilized using fish myofibrillar protein (FMP) at different concentrations (15, 30, and 45 mg mL−1) are investigated. When FMP concentration increases, emulsion activity index decreases (p &lt; 0.05). Conversely, emulsion stability index, adsorbed protein, and interfacial protein contents are augmented (p &lt; 0.05). During storage of 15 days, emulsion added with FMP at 45 mg mL−1 (FMP‐45) has the highest turbidity, zeta potential, viscosity, and viscoelastic properties (G′ and G″) than the others (p &lt; 0.05). Emulsions show shear‐thinning and non‐Newtonian behavior. The emulsion added with FMP at 15 and 30 mg mL−1 (FMP‐15 and FMP‐30, respectively) has dominant viscous properties (G′ &lt; G″), whereas FMP‐45 shows elastic gel‐like behavior (G′ &gt; G″) (p &lt; 0.05). Lower droplet size (d32 and d43), polydispersity index (PDI), flocculation (Fi), and coalescence (Ci) are observe in FMP‐45 (p &lt; 0.05). Microstructure reveals smaller droplet size and smaller increase in droplet size of FMP‐45 after storage. Overall, all emulsions remain stable with no obvious phase separation throughout the storage. Confocal laser scanning microscopic images show that increase in FMP concentration can yield monodisperse emulsion. Therefore, fish myofibrillar protein exhibits excellent emulsifying property and stability for shrimp/soybean oils‐in‐water emulsion.Practical Applications: Oil from shrimp hepatopancreas, a byproduct from hepatopancreas free whole shrimp, is well known for their health benefits. It can be used to produce functional food emulsion such as mayonnaise. The use of fish myofibrillar protein is proven to be a potential emulsifier, which can improve emulsifying properties. The obtained emulsion is stable without no phase separation during the extended storage. This finding can be beneficial for food industries to produce food emulsion rich in high quality protein and polyunsaturated fatty acid. Also, hepatopancreas can be fully exploited.

Packing microstructures and thermal properties of compressed emulsions: Effect of droplet size

In preparing thermodynamically metastable emulsions, the droplets tend to coalesce. Consequently, it is difficult to investigate the properties of a monodisperse compressed emulsion experimentally or by simulations. At a specified volume fraction, the properties of emulsions vary with the droplet size but the relevant studies are very limited. The dissipative particle dynamics simulations are adopted to explore the highly concentrated emulsion of monodisperse droplets. Prior knowledge of the microstructure and inter-droplet interaction is not required. The critical packing associated with the onset of the jammed structure can be identified from the growth of the projected area and perimeter of droplets with the volume fraction (ϕ), ϕc ≈ 0.65. The mean coordination number (Z) from the radial distribution function rises with increasing the volume fraction, and can be described by the scaling relation (Z-Zc) ∼ (ϕ-ϕc)0.82 with Zc ≈ 6.3. The effects of the volume fraction, droplet diameter (D), and interfacial tension (σ) on the internal energy (U) and heat capacity (Cv) are studied systematically. Both U and Cv are found to grow as ϕ and σ are increased but D is decreased. According to dimensional analysis, all the data points can be well represented by the scaling relations (Cv − Cv,c) ∼ ϕ(ϕ − ϕc)1/3(σ/D) and (U − Uc) ∼ ϕ(ϕ − ϕc)1/3(σ/D).