Attractive Van Der Waals Forces Research Articles

Effect of collision on self-assembly of nanoparticles in zirconia microparticle suspension

Nanoparticle halo mechanism is a stabilization method for microparticle suspensions. This study investigates suspension pH and nanoparticles–microparticles collision effects on the stabilization of an aqueous binary suspension. The long-term turbidity measurements show that for the nanosilica suspension stability is directly correlated with pH values; however, in the cases of zirconia and binary suspensions, it is not a monotonic function of pH. It is shown that for binary suspension, the halo mechanism is the primary method affecting the stability of the suspension. The suspension is best-stabilized at pH = 5 that is associated with high halo mechanism efficiency, while increased repulsive forces between micro and nanoparticles at pH = 8 result in halo mechanism decrease and so the stability reaches its minimum. On the other hand, a different trend is observed when the suspension includes only the silica microparticles, maximum stability is at pH = 8, that is related to the higher electrical repulsive forces between microparticles, and it is minimum at pH = 5, because of attractive Van der Waals forces dominancy. Furthermore, the results show that more collision between the nanoparticles and microparticles results in more adsorption of nanoparticles and hence more suspension stabilization at every pH values. This can be attributed to the increased collision between nanoparticles resulting in more nanoparticles be trapped in the profound potential well close to the microparticle surface, and so a stronger nanoparticle halo will form. Besides, it revealed that the effect of the collision on the stability of suspension is more significant at pH = 5.

Anisotropic Assembly of Nanocrystal/Molecular Hierarchical Superlattices Decoding from Tris-Amide Triarylamines Supramolecular Networks.

Directed assembly of nanocrystals from conventional templates suffers from poor control over the periodicity of the nanocrystal assembly, which is largely due to the fact that the template exists prior to the assembly and is not generally adaptive. Herein, small organic molecules (tris-amide triarylamines, TATA) are demonstrated as conceptual templates from self-assembly through noncovalent interactions. The as-formed supramolecular structures with terminated alkyl chains, resembling the structure of as-synthesized nanocrystals capped with alkyl chains, are able to interact with nanocrystals through van der Waals attractive forces, thereby enabling directed assembly of nanocrystals into ordered superlattices. Specifically, it is found that, as determined by the substituted alkyl chains of TATA, either H or J-aggregates of TATA can be achieved, which eventually produce several distinct supramolecular structures, from rods to spindles, to rings, and to spheres, serving as on-pathway intermediate that directs the assembly of nanocrystals into diverse nanocrystal superlattices. The approach described can be applicable to produce ordered nanocrystal assemblies of a wide range of nanocrystals, independent of size and shape and without ligand exchange process. Strikingly, a helical TATA stacking can direct assembly of binary nanocrystal mixtures into NaZn13 binary superhelix.

Determination of Hamaker constant for CNT and AFM probe configuration inside SEM

Mounting the atomic force microscope (AFM) probe on the gripper of the robotic nanoscale manipulators system enables the combination of the AFM and scanning electron microscope (SEM). The AFM–SEM-based method of determining Hamaker constant for carbon nanotube (CNT) and AFM cantilever in the vacuum condition of SEM, which is completed through a robotic nanoscale manipulators system, is presented for the first time. During Hamaker constant determining experiment, the elastic force is measured utilising integrated AFM–SEM imaging technique through detecting the deflection of AFM cantilever that actually equals van der Waals adhesion force. The van der Waals attraction force between CNT and AFM cantilever is calculated through the modelling of a cylindrical nanotube and a flat gold-coated surface. The experiment results indicate that Hamaker's constant measured are comparable to typical values published by previous researchers, being of the order of 100 zJ, which indicates that the present approach is valid and reliable.

Stability of nanobubbles in different salts solutions

The stability of nanobubbles in electrolyte solutions under different ion valence values was studied using deionized water, NaCl, Na2SO4, Na3PO4, CaCl2, and FeCl3. Nanobubbles were generated using hydrodynamic cavitation, and bubbles were tested for size and zeta potential. All the samples were stable for one week with no significant deviation in either bubble size or zeta potential values. The variation of size and zeta potential among six samples can be attributed to the solution properties and was mainly dependent on solution pH and the cation valency. The ion profiles revealed that the cation concentration at the bubble surface was higher than that of bulk, confirming that the bubbles were negatively charged for neutral and high pH values (≥4) under low valency cation adsorption. The high valency cations have the potential to neutralize or completely reverse the bubble charge. Anions or co-ions have minimal effect on the surface potential or the surface charge. The calculated internal pressures of bubble were unrealistically high, suggesting that the surface tension should be lower than that of water for nanobubble solutions. The interaction energy profile shows no significant energy barrier that overcomes the attractive van der Waals forces for all the solutions, except NaCl which had a 1.87 × 10−20 J barrier at a 5 nm separation distance. However, with the recorded stable bubbles, the calculation of the attractive van der Waals forces produced unrealistic values indicating that the Hamaker constant used for the calculation may not be valid at the nanobubble gas-liquid interface. This revealed that nanobubbles should contain exceptional interfacial properties that need to be carefully investigated and evaluated.

Graphite/n-InSe photosensitive junctions fabricated by transferring dry drawn graphite film onto as-cleaved surface of InSe

We report on reproducible fabrication and characterization of new photosensitive Schottky-type junctions prepared by an extremely simple technique when a substrate-free dry drawn graphite film is transferred from water surface onto as-cleaved van der Waals surface (0001) of n-lnSe substrate. Electrical and photoelectric characterization of the obtained junctions is investigated using current-voltage, capacitance-voltage and photosensitivity spectra measurements. It is revealed that the thin graphite film transferred onto the n-lnSe (0001) substrate creates a Schottky junction with a potential barrier of 0.49 eV and rectification ratio of RR ~ 200 at 0.9 V. An abrupt rectifying electrical junction graphite/n-lnSe is formed due to van der Waals attraction forces between the thin graphite film and atomically smooth n-lnSe (0001) surface. The graphite/n-lnSe junctions generated short-circuit current density of Jsc = 0.29 mA/cm2, open-circuit voltage of Voc = 0.31 V and filling factor of FF = 0.33 under 80 mW/cm2 white light illumination. The junctions exhibit photosensitivity in the near-infrared and visible spectral ranges, making them promising for photodetector applications. Their energy band diagram is built up.

Sidewall hydrogenation impact on the structure and wettability of spaghetti MWCNTs

Carbonaceous and hydrophobic surfaces of carbon nanotubes require modification by functionalization to interact more efficiently with the surrounding medium such as air, water and bio-fluids. Two types of spaghetti MWCNTs, a less studied category of nanotubes, with the advantage of simpler incorporation with other materials, were fabricated, characterized and hydrogenated. The structural differences of the spaghetti types, mainly due to the structural differences of the substrates they were grown on, were reflected in FESEM and TEM images, XRD patterns and Raman spectra of the two samples. More interestingly, water contact angle measurements on initial hydrophobic structures showed a distinct wettability behavior in the two types with hydrogenation time, where one became even more hydrophobic and the other switched to hydrophilic. The inherent structural differences between the two types, the surface roughness, the increase in the hydrogen content on the sidewalls of the nanotubes and the interplay of repulsive Columbic and attractive van der Waals forces among the hydrogens and the nanotubes may have all played a role in this wettability behavior. However, the capability of switching surface wettability on demand is very useful for future environmental applications and may support unique devices in microfluidics, catalysis, and detection systems.

Signature of Short-Range van der Waals Forces Observed in Poisson Spot Diffraction with Indium Atoms.

The phase of de Broglie matter waves is a sensitive probe for small forces. In particular, the attractive van der Waals force experienced by polarizable atoms in the close vicinity of neutral surfaces is of importance in nanoscale systems. It results in a phase shift that can be observed in matter-wave diffraction experiments. Here, we observe Poisson spot diffraction of indium atoms at submillimeter distances behind spherical submicron silicon dioxide particles to probe the dispersion forces between atoms and the particle surfaces. We compare the measured relative intensity of Poisson's spot to theoretical results derived from first principles in an earlier communication and find a clear signature of the atom-surface interaction.

Silica anchored colloidal suspension of magnetite nanorods

This study focuses on the interaction of colloidal silica nanoparticles with the magnetite (Fe3O4) magnetic fluids (MF), which eventually forms nanorod like structure. The aqueous magnetic fluid consists of magnetite nanoparticles having double layers of lauric acid surfactant. This surfactant provides stability towards short-range van der Waals attractive and steric repulsive forces, as well as long-range dipole-dipole interactive force. Whereas, in the colloidal silica, the sodium ions provide stability to the silica nanoparticles. The colloidal silica and magnetic fluid both were mixed in different proportions to understand the interaction between the silica and magnetite nanoparticles. Thus, the interaction present in the system is studied using FTIR, TGA, and magnetic field induced microscopy. The FTIR and TGA data reveal that silica interacts with the outer layer of lauric acid through the Si–O bond and eventually provides stability to the system. The length of the lauric acid sheath varies with the concentration of silica nanoparticles. The SEM images indicate nanorod formation, and its structure dimensions vary with the silica concentrations, which is also reflected in the magnetic field-induced structure formations. The structure observed using microscopy correlated with the interaction derived from FTIR and TGA data analysis. XRD data of iron oxide (magnetite) nanoparticles are present here again for completion.

Colloidal forces, microstructure and thixotropy of sodium montmorillonite (SWy-2) gels: Roles of electrostatic and van der Waals forces

A model for explaining the effect of ionic strength on the yield stress and thixotropic behaviour of sodium montmorillonite or NaMt (SWy-2) gel was presented. NaMt gels displayed thixotropic behaviour at low to 0.1 M KCl concentration. This behaviour is reflected by an increasing yield stress with rest time after preshearing. The increasing yield stress measured the strengthing of the gel structure as it recovered at rest. The microstructure was formed by highly flexible nanosized platelets interacting both attractively via heterogeneous charge attraction in the edge-face and overlapping edge-edge configurations, and repulsively between the faces via electric double layer (EDL) force. This EDL force opened up the microstructure and was responsible for both the strength and thixotropic behaviour of the gel. The increasing gel strength with ionic strength up to 0.1 M KCl was a result of the weakening of the EDL force between the faces (interacting at an angle) particularly at the platelet bond forming the network junctions. The heterogeneous charge attraction forming the bond effectively increased in strength. In addition, weak platelet bonds were also able to form. The EDL force governs the microstructure development and this process takes a long time for every platelet to experience a minimum net force. The Leong model described the ageing or thixotropic behaviour quite well. At higher KCl concentrations, the van der Waals attractive force dominated the platelet-platelet interactions. These suspensions displayed time independent behaviour and was found to belong to the flocculated phase state.

Antimicrobial Properties of Polyaniline and Polypyrrole Decorated with Zinc-Doped Copper Oxide Microparticles.

Polyaniline (PANI) and polypyrrole (PPY) were synthesized by carbon dots (CDs) under UV irradiation and then sonicated together with zinc acetate and copper acetate to form the PANI-Zn@CuO and PPY-Zn@Cu composites. The former consisted of agglomerated spherical particles with diameters of 1–5 µm, whereas the latter displayed irregular stick shapes with similar diameters. The bacterial potency of the composites against Escherichia coli and Staphylococcus aureus was enhanced remarkably with Zn doping in the CuO matrix, designated as Zn0.11Cu0.89O, at 0.144 mg/mL. The cell death was mainly attributed to the release of reactive oxygen species (ROS) that would severely damage DNA, proteins, and lipids. Bacteria could adhere to neutral surfaces of the composites by van der Waals attractive forces. The binding event disrupted the native surface charge of bacterial cells to induce cell lysis and result in eventual cell death.

Salt Enhances the Thermostability of Enteroviruses by Stabilizing Capsid Protein Interfaces.

Enteroviruses are common agents of infectious disease that are spread by the fecal-oral route. They are readily inactivated by mild heat, which causes the viral capsid to disintegrate or undergo conformational change. While beneficial for the thermal treatment of food or water, this heat sensitivity poses challenges for the stability of enterovirus vaccines. The thermostability of an enterovirus can be modulated by the composition of the suspending matrix, though the effects of the matrix on virus stability are not understood. Here, we determined the thermostability of four enterovirus strains in solutions with various concentrations of NaCl and different pH values. The experimental findings were combined with molecular modeling of the protein interaction forces at the pentamer and the protomer interfaces of the viral capsids. While pH only had a modest effect on thermostability, increasing NaCl concentrations raised the breakpoint temperatures of all viruses tested by up to 20°C. This breakpoint shift could be explained by an enhancement of the van der Waals attraction forces at the two protein interfaces. In comparison, the (net repulsive) electrostatic interactions were less affected by NaCl. Depending on the interface considered, the breakpoint temperature shifted by 7.5 or 5.6°C per 100-kcal/(mol·Å) increase in protein interaction force.IMPORTANCE The genus Enterovirus encompasses important contaminants of water and food (e.g., coxsackieviruses), as well as viruses of acute public health concern (e.g., poliovirus). Depending on the properties of the surrounding matrix, enteroviruses exhibit different sensitivities to heat, which in turn influences their persistence in the environment, during food treatment, and during vaccine storage. Here, we determined the effect of NaCl and pH on the heat stability of different enteroviruses and related the observed effects to changes in protein interaction forces in the viral capsid. We demonstrate that NaCl renders enteroviruses thermotolerant and that this effect stems from an increase in van der Waals forces at different protein subunits in the viral capsid. This work sheds light on the mechanism by which salt enhances virus stability.

A Simple Method to Determine Critical Coagulation Concentration from Electrophoretic Mobility

Critical coagulation concentration (CCC) is a key parameter of particle dispersions, since it provides the threshold limit of electrolyte concentrations, above which the dispersions are destabilized due to rapid particle aggregation. A computational method is proposed to predict CCC values using solely electrophoretic mobility data without the need to measure aggregation rates of the particles. The model relies on the DLVO theory; contributions from repulsive double-layer forces and attractive van der Waals forces are included. Comparison between the calculated and previously reported experimental CCC data for the same particles shows that the method performs well in the presence of mono and multivalent electrolytes provided DLVO interparticle forces are dominant. The method is validated for particles of various compositions, shapes, and sizes.

Porous composites based on cellulose acetate and alfa-hematite with optical and antimicrobial properties

Polymer nanoparticles (NPs) composites are rapidly gaining attention due to their high potential for application in several technological fields. On one hand, polymers contribute to balance the attractive magnetic and van der Waals forces that act on NPs and thus reduce their agglomeration. On the other hand, NPs can bring into the composites their inherent properties as nanosized objects. In this work, hematite NPs were blended with cellulose acetate (CA) polymer and thin nanocomposite films were produced by the non-solvent induced phase separation technique. A full physicochemical characterization of these composite materials confirmed its superior character in terms of structure, thermal stability, electronic/optical and antibacterial properties when compared to the polymer on its own and opens the door to its further validation in adsorptive and photocatalytic applications.

Predicting the interaction between nanoparticles in shear flow using lattice Boltzmann method and Derjaguin–Landau–Verwey–Overbeek (DLVO) theory

The functionality and performance of colloidal suspensions used in catalyst layer preparation and biomedical applications are largely dependent on the interaction between nanoparticles in colloidal suspension systems. Previous models (e.g., collision model) usually rely on an artificial repulsive force as the sole interaction between nanoparticles to prevent overlapping, but fail to capture the agglomeration or reveal the effect of solvents. In this study, the Derjaguin–Landau–Verwey–Overbeek (DLVO) theory is implemented in conjunction with a lattice Boltzmann-smoothed profile method developed to simulate the dynamic solid–fluid and particle–particle interactions between nanoparticles in shear flow. Both aqueous and non-aqueous solvents are considered. The model consists of an attractive van der Waals force and repulsive electrostatic and Born forces in aqueous solvents and is modified for non-aqueous solvents by replacing the repulsive electrostatic force by Coulombic repulsion. The numerical model is validated against a benchmark analytic solution for the motion of one nanoparticle in shear flow. For two-particle systems, physically representative simulations are obtained with the DLVO models, resulting in nanoparticles that remain attached or eventually detach depending on a critical particle Reynolds number. Furthermore, the DLVO models properly resolve the effect of solvents on nanoparticle motion. The improved representation of inter-particle interactions achieved with the DLVO and modified-DLVO models provides a physically consistent approach to simulate and investigate agglomeration and dispersion in colloidal suspensions.

Importance of soil interparticle forces and organic matter for aggregate stability in a temperate soil and a subtropical soil

Soil organic matter (SOM) and interparticle forces, including the electrostatic repulsive force (ERF), surface-hydration repulsive force (SHRF) and van der Waals attractive force (vDWAF), play crucial roles in aggregate stability. However, few studies investigated their coupled effects on aggregate stability in variably charged soils, in which soil particle interactions are more complex than in permanently charged soils due to the coexistence of positive and negative charges and the variable surface charge characteristics. Therefore, this study aims to: 1) investigate the combined effects of soil solution pH and the overlapping of electric double layers (EDLs) between positively charged Fe/Al (hydro)oxides and negatively charged SOM on the ERF and aggregate stability before (control) and after SOM removal and after straw incubation for 240 days; and 2) evaluate the importance of soil interparticle forces and SOM for aggregate stability in a permanently charged temperate soil (Vertisol) and a variably charged subtropical soil (Ultisol). Soil aggregate stability was determined by measuring the release of small particles (w(<d)%) after aggregate breakdown at different KCl concentrations (10−1 to 10−5 mol L−1). Soil solution pH was adjusted by varying KCl concentration. In both soils, the w(<d)% increased exponentially with the net force of the soil interparticle forces. However, in contrast to the Vertisol, the w(<d)% of the differently treated Ultisol showed almost no change before the KCl concentration decreased to 10−2 mol L−1 and continuous changes after the KCl concentration decreased to 10−2 mol L−1. In addition, the differences in the w(<d)% between the SOM removal treatment and the control and straw incubation treatments were much larger in the Ultisol than in the Vertisol. In the Ultisol, the ERF did exhibit little and continuous increase over the whole range of tested pH. When considering the impact of the overlapping of oppositely charged EDLs between Fe/Al (hydro)oxides and SOM, a much stronger reduction in ERF was observed in the control and straw incubation treatments and at lower KCl concentrations. Consequently, in both the Vertisol and Ultisol, aggregate stability is essentially controlled by the soil interparticle forces, whereas the effects of soil solution pH and the overlapping of oppositely charged EDLs between SOM and Fe/Al (hydro)oxides on the ERF must be considered in variably charged soils (i.e. Ultisol). Moreover, SOM can play a more important role in aggregate stability in subtropical soil than in temperate soil.

Coagulation behavior of spherical particles embedded in laminar shear flow in presence of DLVO-and non-DLVO forces

HypothesisComplex and coupled interaction between coagulation mechanisms results in a nonlinear variation of coagulation rate with shear rate. CalculationsCoagulation behavior of colloidal dispersions is investigated in a laminar shear flow by solving the Fokker-Plank equation for pair probability density function, simultaneously incorporating the effect of (i) Brownian diffusion, (ii) fluid flow, (iii) van der Waals attraction and (iv) double layer repulsion force. Furthermore, analysis foremost studies the effect of non-DLVO solvation force. FindingsTheoretical analysis with experimental validation reveals that, coagulation rate varies non-linearly with the shear rate in presence of double-layer repulsion force, due to the strong coupling of coagulation mechanisms. This is observed by an occurrence of coagulation minima at intermediate shear rate. Increase in double layer repulsion force either facilitates or hinders the occurrence of redistribution of particles, as observed by early or late fall of coagulation rate. Systems with different ionic strengths show a crossover of coagulation rates, whereas those with different solvation forces do not show similar trend. Analysis also shows that, ad hoc additivity assumption of individual Brownian and shear coagulation rates does not hold in case of laminar shear flow, against to that observed in extensional flow [Melis et al. (AIChE J., 999, 1383)].

Rheological properties for fresh cement paste from colloidal suspension to the three-element Kelvin–Voigt model

In the present work, rheological behaviors of fresh cement paste are studied based on multi-disciplinary approaches, i.e., colloidal suspension using attractive van der Waals force, rheology using the Bingham model and Bingham–Papanastasiou model, which are able to describe the behavior of cement pastes before and after yield stress, and finally the continuum mechanics based on Maxwell and Kelvin–Voigt models. To achieve this, the fresh cement paste with different water-to-cement ratios of 0.3 up to 0.6 is prepared. The attractive van der Waals forces are estimated based on the distances between solid cement particles, which vary at every single water-to-cement ratio. The rheology experiments of all water-to-cement ratios are performed using a rheometer. According to our experimental outcomes, the Bingham and Bingham–Papanastasiou models are applied in the modeling of the experimental curves and determination of yield stress and viscosity. Maxwell and the Kelvin–Voigt models are utilized in describing solid-like behavior before yield stress and fluid-like behavior beyond yield stress. It is observed that the increase of water generates a decrease in the viscosity, yield stress, and packing concentration of solids. It also increases the distances between two cement particles in the cement pastes. According to the modeling results, the Bingham–Papanastasiou model is well adapted for the cement paste flow due to its additional modeling parameter, which is known as m. The role of m is understood and described by linking the van der Waals interaction, rheology, and three-element Kelvin–Voigt model as a whole in function of water-to-cement ratio. m is understood as a key parameter in which the distance between particles affects the rheological behavior of fresh cement pastes. Lastly, the two-phase flow simulations have been successfully achieved and compared with the experiments. The conclusion and outlooks are summarized and discussed at the end of the paper.

Shear rate dependence of viscosity and normal stress differences in ferrofluids

In this work we investigate numerically the viscosity and normal stress differences of mono and polydisperse ferrofluids undergoing a simple shear flow in the presence of a transverse applied magnetic field. The simulations are carried out by means of a Brownian Dynamics integration algorithm. For all simulations we assume that the dipole moments are fixed to the magnetic particles. In the case of polydisperse suspensions the particles present a typical log-normal distribution of magnetic diameters which are covered with a layer of surfactant of constant width. The long-range dipolar interaction between the magnetic moments are computed by Ewald summation technique and the surfactant influence is modeled as a steric repulsive force between pairs of particles. The attractive van der Waals force is also incorporated in the numerical simulations. The numerical procedure is validated by comparing the simulation results of the non-dimensional relative viscosity increase as a function of the magnetic field intensity with experimental data. The present numerical simulations show that the non-dimensional relative viscosity increment as a function of the non-dimensional shear rate is strongly influenced by the particle volume fraction and the strength of the dipolar interactions parameter. By examining the suspension microstructure into the numerical box, we can see anisotropic structures like chains of particles for the low shear rate regimes, specially in the polydisperse cases. The results also indicate the presence of a suspension shear-thinning behavior with two different non-dimensional shear rate dependence. In addition, non-dimensional normal stress differences are observed even in the absence of dipolar interactions, as a consequence of stress anisotropy produced by the repulsive forces during the particle collisions induced by the imposed steady-shear flow. The dipolar interactions increase these non-dimensional normal stress differences and make their dependence on the non-dimensional shear rate non-monotonic. For low non-dimensional shear rates, the dipolar interactions dominate and the formation of large-structures like chains oriented in the direction of the magnetic fields leads to an increase in the non-dimensional first normal stress difference. For very high non-dimensional shear rates, the flow breaks the anisotropic structures and the effects of polydispersity and dipolar interactions become unimportant. The numerical simulations provide a useful tool for understanding the complex microphysical and rheology of real ferrofuids under shear flows.

Formation Mechanism of ZnO Nanorod Superstructures in Different Solvents

AbstractHierarchical superstructures consisting of nanoscale building blocks show great potential in achieving configurable properties. In this work, two patterns of ZnO nanorod superstructures are obtained directly by thermal decomposition of zinc acetylacetonate in different solvents. Namely, fan‐shaped structures are obtained in methanol, whereas side‐by‐side assembly is achieved in n‐pentanol. Time‐dependent observations indicate the formation of quasi‐spherical particles at the beginning of the reaction, which later transform into different superstructures depending on the types of solvents. The selective formation of superstructures is rationalized based on the calculation of interaction potentials between two nanorods with either crossed or side‐by‐side spatial organizations. The results reveal the significant role of solvents in affecting electrostatic repulsion, which may compete with attractive van der Waals forces leading to the favored superstructures.

Adaptive Control of Atomic Force Microscope for Surface-Profile Estimation

This paper describes a methodology for designing an adaptive tracking controller of an atomic force microscope to estimate a surface profile of a sample. A microbeam of the atomic force microscope is modeled as an Euler-Bernoulli beam with single mode considered. A tip-sample interaction force used here is a piecewise function described by the attractive van der Waals force and the repulsive Derjaguin-Muller-Toporov force which can represent the indentation made by a tip of the microbeam into the sample surface. The adaptive-tracking controllers for both regions are designed based on adaptive control Lyapunov functions. With the driving and measured signals, the controlled acceleration is generated and superimposed onto the harmonic excitation and the estimated surface profile is obtained. If the tip of the microbeam taps harder on the sample surface, the more estimation error is obtained. To reduce the estimated surface-profile error, the regression model needs more higher-order terms to approximate the repulsive Derjaguin-Muller-Toporov force.