A new educational approach to surface tension and Young-Laplace law by using computer-based simulations

Here we present a pedagogical approach to surface tension phenomena based on a novel mesoscopic fluid model. We show that computer simulations based on this model can provide quantitative results that are consistent with experimental results. We simulate the formation of a two-dimensional liquid droplet without gravity and obtain its surface tension as a result by also verifying Young-Laplace’s law. By controlling some simulation parameters, students can understand step by step how some physical quantities (e.g., the intensity of the interaction between two fluid particles) can affect the behaviour of fluids. This approach will provide students with an explicative model of surface phenomena.


Introduction
Traditional pedagogical approaches to surface phenomena often rely on a macroscopic description of the forces acting at interfaces and on a microscopic interpretation.Apart from sometimes being incorrect or contradictory, both approaches often prove ineffective in engaging students and fostering authentic understanding of physics content [1].Our research aims to develop and test alternative strategies to improve the teaching and learning of surface phenomena, including the use of novel mesoscopic fluid models.In the scholarly literature, mesoscopic models are widely acknowledged for their efficacy in educational settings, particularly in facilitating the comprehension of fundamental concepts like surface tension.These models have the advantage over microscopic models in that they can explain surface tension using interparticle interactions, although they do not account for thermal fluctuations.In addition, mesoscopic models do not require many computational resources to successfully run the simulations.Therefore, teachers/students can even simulate large portions of fluids using computers commonly found in school laboratories.We present here a mesoscopic fluid model and some computer simulation results based on this model.The simulations have been intentionally constructed to provide students with the ability to manipulate certain pertinent parameters within the model.We simulate the formation of a two-dimensional liquid droplet without gravity and determine its surface tension by also verifying Young-Laplace's law.This approach should allow students to become familiar with surface phenomena.By controlling some simulation parameters, students can understand step by step how some physical quantities (e.g., the Smoothed particle hydrodynamics (SPH) is a computational technique that employs a Lagrangian approach to approximate solutions for the equations governing fluid dynamics.In this method, the fluid is represented by a collection of particles.The SPH method involves the assignment of physical properties, such as mass and density, to individual particles representing the fluid.The physical quantities are acquired through the process of interpolation.
The expression of a continuous quantity Q(r) as a function of position r can be mathematically represented by the following convolution integral.
where ( −  ′ ) is the Dirac Delta function.
In the SPH method, the Dirac Delta function is approximated using weighting functions, commonly referred to as W, in order to represent a generic smoothed quantity   () = � ( ′ )( −  ′ , ) ′ . ( The weighting functions, denoted as W, play a crucial role in characterising the SPH method.These functions are defined within the range of H, which is also referred to as the "smoothing length".The variable ri denotes the spatial location of the ith particle. The interpolation integral in eq. ( 2) can be expressed as where ( ′ ) ′ is a small portion of mass.Hence, in order to calculate the physical quantity associated to a generic particle i, the integral in eq. ( 2) is estimated through the subsequent summation [4].
where mj, Δ  and   are the mass, the volume and the density of the particle j, respectively, and are functions of the position   .The summation in equation (3) includes all particles in principle.However, the influence of the particle i is primarily limited to its immediate neighbouring particles, as the function W exhibits a rapid decrease in magnitude as the distance between particles increases.
The gradients of these quantities can also be approximated through a summation.The study of particle motion and its properties can be conducted through the utilisation of SPH formulations, which involve the replacement of quantities and their gradients in the Navier-Stokes equations.Numerous formulations of the Navier-Stokes equations using SPH have been documented in the existing academic literature.[4].The present study employs the following momentum-conservation equation, [5] where  is the dynamic viscosity,   and Pi indicate the velocity and the pressure of the generic particle i, respectively.The initial term on the right-hand side, which involves the SPH formulation of the pressure gradient, was derived by Lucy [3] and represents the acceleration resulting from the pressure force.The acceleration due to the SPH formulation of the viscous force, as derived by Morris et al. [6], is represented by the second term on the right-hand side.In the present model, the liquid exhibits weak compressibility, as this characteristic enhances the stability of the numerical evolution over time [7].The pressure in equation ( 4) can be calculated by considering a state equation.The following Tait equation is employed for the determination of liquid pressure [8].
where c0 is the sound speed,  is the density of the liquid and  0 a reference density (in our analysis this value is fixed to 1000, in arbitrary units).The sound speed was established to be 50 times higher than the maximum velocity achievable by an SPH particle.This selection guarantees that variations in density can be regarded as insignificant [7].The simulation incorporates the concept of "artificial viscosity" [7], a commonly employed technique in SPH algorithms to mitigate instabilities during the simulation process.The selection of the "artificial viscosity" parameter must be executed with great precision in order to achieve accurate estimations of the pressure within the droplet.
In order to accurately replicate surface tension phenomena, the model incorporated an additive force Fij that considers the interactions between liquid particles and solid boundaries.Given the inherent relationship between surface tension and inter-molecular interactions, it is justifiable to incorporate an interaction force into the mesoscopic model that exhibits repulsion at close distances and attraction at greater distances.This force enabled the characterization of capillary phenomena and the examination of the interactions between liquid and solid particles that give rise to capillarity [9].The magnitude of this force is contingent upon the separation between particles.In order to optimise computational efficiency and minimise the computational burden associated with inter-particle interactions, it is imperative to implement a cut-off on the range of these interactions.In the present investigation, the established threshold is defined as a distance equivalent to twice the value of H.The force referred to as the "molecular-like" force, as described by Tartakovsky and Meakin [5], is determined through the utilisation of a specific mathematical function reported below.
Sij and |rj-ri| indicate the magnitude of the force between two particles i and j and the distance between them, respectively.This mathematical expression is similar to the one used by Tartakovsky and Meakin but with a support equal to 2H and not H.The force Fij exhibits antisymmetry, denoted by Fij = -Fji, in accordance with the principle of momentum conservation.In the context of inter-particle interactions, it is observed that the force between particles is repulsive when the distance between them is less than 2⁄3 times the characteristic length scale denoted as H. Conversely, when the inter-particle distance falls within the range of [2⁄3 H, 2H], the force becomes attractive.Finally, when the inter-particle distance surpasses 2H, the force between the particles becomes zero.
The significance of the short-range repulsive force in achieving liquid behaviour consistent with empirical observations has been duly acknowledged, as evidenced by numerous research findings [10,11,12].
In order to accurately calculate the total force exerted on a particle by all other liquid particles, the momentum equation is expressed as follows.
Fi is the whole force exerted on a particle i and calculated by using equation (6).
In this study, the researchers utilise a smoothing kernel function known as Wendland [7].The polynomial function exhibits favourable characteristics in terms of resistance to tensile instability [13].The calculation of time evolution is performed using a second-order method [7], which is a modified version of the leapfrog scheme [7].

Droplet pressure vs radius
The pressure at mechanical equilibrium reached within a liquid droplet in the absence of gravitational forces is determined by the surface tension and the radius of the droplet, as described by the Young-Laplace law.At the onset of the simulation, the SPH particles are uniformly distributed in a rectangular configuration, with a separation distance of ds between adjacent particles.The experimental outcomes were acquired by establishing the following parameters: H = 0.48 mm, ds = 0.16 mm as the initial interparticle distance, and Sij = -1 × 10 −5 a.u. as the liquid-liquid interaction force.Figure 1 illustrates the interparticle force, denoted as Fij, based on the aforementioned parameters.A series of numerical experiments were conducted to investigate the formation of a two-dimensional liquid droplet in the absence of gravity.The experiments focused on observing the droplet formation as a function of its radius, and subsequently determining the corresponding surface tension values for each radius.Figure 2 illustrates an instance of a droplet in a state of static equilibrium, with emphasis placed on the molecularlike and pressure forces.At the point of equilibrium, the simulation yields two key parameters: the pressure within the droplet and its corresponding radius.
The determination of surface tension involves the consideration of the widely recognised Young-Laplace equation in the context of two-dimensional systems, as reported below.
σ is the surface tension, R the radius and PT the pressure inside the drop.
At equilibrium, the pressure within the droplet is contingent upon the cumulative force exerted on each individual particle.According to Hoover (2014), it has been demonstrated SPH exhibits isomorphism with molecular dynamics, specifically in the context of particle-particle interactions involving multiple bodies.
The utilisation of the SPH equations and the consideration of particle-particle interactions in a coherent manner enable the calculation of pressure based on the cumulative forces resulting from such interactions where Pk is the ideal gas (kinetic) contribution to the pressure, r is the "virial" radius of a circle inside the liquid drop and   is the interaction force.In the scenario of a drop in a state of equilibrium, the value of Pk The summation includes all the particles located within the virial radius r (identified by the index i) and all the particles contained within the drop (identified by the index j).Exclusions are made for self-interactions.
Figure 3 illustrates the values of the pressure PT, obtained through the utilisation of equation ( 9), in relation to the reciprocal of the radius R for various droplets of distinct sizes.Additionally, a linear regression analysis has been conducted employing the Young-Laplace law.Through the process of linear fitting, it is possible to make an estimation of the surface tension value (represented by the symbol σ) for the simulated liquid.The obtained value for the standard deviation, σ, is 0.05064 ± 0.00004 atomic units (a.u.).The coefficient of determination is approximately 0.99.

Discussion and Conclusions
The success of model-based approaches in physics learning has long been acknowledged in the literature (see, for example, [15]).Gilbert et al. recognised model creation in particular as a superordinate process skill [16] that can encourage the learner's critical thinking abilities.According to other research, including modelling activities in physics classes might help students develop a positive attitude towards learning the subject, make it easier for them to understand it, and help them recognise similarities and differences across a variety of occurrences [17,18,19].
According to research by Rutten et al., interactive simulations are an effective way to introduce modelling to students.It has been shown to be beneficial in generating learning that is meaningful for the learners to give students the opportunity to actively change model parameters and observe the results in real-time [20,21].More specifically, the ability to solve problems and make decisions is crucial in simulation exercises.The pupils may have the chance to concentrate on the mechanism of functioning behind the observable phenomena by step-by-step dissecting how the parameters connected to inter-particle interactions affect liquid behaviour.Students can learn how to use a model to anticipate results starting with tested hypotheses and gain an understanding of the value of the model building process through the use of simulations.
The simulation exercises that we developed are just one component of a larger teaching and learning sequence that also features significant hands-on, minds-on experimental activities as well as interactive courses that are based on active learning methodologies.Our sequence was tested with a defined group of high school students and the teachers who accompanied them; the findings of some of the pertinent components of that research are detailed elsewhere [22,23].

Figure 1 .
Figure 1.The dependence of the force per unit mass on interparticle distance rij.

Figure 2 .
Figure 2.An illustration of a droplet in the state of mechanical equilibrium, derived from the outcomes of a simulation conducted across multiple time steps.Both the molecular-like and pressure forces are highlighted.

Figure 3 .
Figure 3. Droplet pressure values as a function of the inverse of the droplet radius and the linear regression.
is equal to zero.Similarly, the absence of viscous forces can be attributed to this condition.Therefore, the forces of interaction are   =     �  �  −   , � +