Phase Field Modeling - Science topic

In multiphase flow modeling, where you're simulating interactions between different phases (like liquid and gas), you need a way to track or represent the interface—the boundary between the phases. There are two main approaches: interface tracking and interface capturing.

1. Interface Tracking Methods

What it means: Interface tracking methods focus on directly following the interface between the phases. Think of it like trying to keep track of the edge of a soap bubble as it moves and changes shape over time. In this approach, the interface is treated as a well-defined line or boundary that you actively track during the simulation.

These methods use markers or particles that follow the interface. For example, in a "marker-and-cell" method, you'd use special particles to mark the location of the interface and track them as they move with the flow of the fluid.

- Pros: The interface is well-defined and sharp, so you get very precise information about where the two phases meet.

- Cons: It can get computationally expensive, especially when the interface undergoes complex changes (like merging, splitting, or highly irregular shapes). Special treatment is often required for such changes.

2. Interface Capturing Methods

- What it means: In interface capturing you're not directly tracking the interface. Instead, you use a smooth, continuous field that changes across the interface, and this field captures the boundary between phases. The interface is implicitly defined by where the field changes value.

For example, you might use a "phase indicator" field that smoothly transitions from one phase to another. The interface is located where this field crosses a certain value, like zero. The field doesn’t have to directly follow the interface, but it still captures its position.

In the Level Set method, a function called a signed distance function is used, and the intrface is defined where this function equals zero. In the Phase Field method, an order parameter smoothly changes across the interface, and the interface exists where this change happens.

- Pros: These methods can easily handle situations where the interface changes shape, splits, or merges, which can be tricky for tracking methods. They are typically easier to implement for more complex flows.

- Cons: The interface might not stay as sharp as in tracking methods. Instead, it may become "diffuse," meaning the boundary is less clearly defined. Some extra steps are usually needed to keep the interface accurate.

Where do methods like Phase Field and Level Set fit in?

- Phase Field Method: This is an interface capturing method. It uses a continuous field (called the order parameter) to represent different phases, and the interface is where this field transitions between them. The boundary is not tracked directly, but instead captured by the way the field behaves.

- Level Set Method: This is also an interface capturing method. It uses a signed distance function to describe the position of the interface. The interface is implicitly defined as the point where the function equals zero, and the function evolves over time to capture changes in the interface.

all of them gives very good results and I'm focusing on XFEM in abaqus software which gives very accurate results specially for complex geometries. I have already explain it in this 3-hour course here

Simulate Three-Phase Flow with a New Phase Field Interface

📷

by Ed Fontes

December 16, 2015

In COMSOL Multiphysics version 5.2, the CFD and Microfluidics modules include a new fluid flow interface for modeling separated three-phase flow. The model behind this fluid flow interface accounts for surface tension between each pair of fluids, contact angles with the walls, as well as the density and viscosity of each of the fluids. The phase field method computes the shape of the interfaces between the three phases and also accounts for interactions with walls.

Modeling Separated Multiphase Flow

COMSOL Multiphysics has offered modeling and simulation capabilities for multiphase flow for many years. However, the equation-based formulation of three-phase flow problems has been used successfully by only a few experts in our user community, as seen in the figure below. In the last two or three years, we have received numerous requests for a more user-friendly, ready-made three-phase flow interface. In COMSOL Multiphysics version 5.2, we have satisfied these requests.

📷 Simulation results of a rotating drum with three-phase flow, performed by a COMSOL Multiphysics user with equation-based modeling.

The description used in this new flow interface, the Three-Phase Flow, Phase Field interface, is a separated multiphase flow model. This interface is similar to the level set and phase field models for the two-phase flow interfaces included in earlier versions of the software. This means that the interface between the three immiscible phases (air, oil, and water, for example) is resolved in detail, including the effects of surface tension and contact angles.

Separated multiphase flow models are used for studying microfluidic systems and for fundamental studies of bubble coalescence and droplet breakup mechanisms. These are processes and phenomena where surface tension, contact angles, and buoyancy effects play a significant role on the shape of the phase boundaries between the different fluids and on the velocity field. Microfluidic devices and processes are typically found in analytical chemistry, biotechnology, medical technology, and nanotechnology. They include inkjets, sensors, separation devices, lab-on-a-chip devices, and microreactors.

High-fidelity separated multiphase flow models are usually too computationally expensive for direct use in macroscopic descriptions, since that may require resolving the surface of thousands or millions of droplets and bubbles. However, the results of detailed fundamental studies on a few bubbles and droplets can be used to develop simplified, less computationally expensive models. These simplified descriptions can usually be included in macroscopic dispersed multiphase flow models, which can describe systems with millions of bubbles and droplets. Macroscopic dispersed multiphase flow models are interesting in the study and design of devices and processes in the pharmaceutical, food, chemical, and household product industries.

The tutorial included in the Application Library, shown in the image below, deals with a droplet of air that rises through a layer of water at the bottom of a container and then into a layer of oil, lighter than water, resting on the water’s surface. As the air bubble moves through the water-oil interface, it carries some water in its wake and into the layer of oil. The water entrained in the bubble’s wake forms a “water tail” behind the bubble in the oil layer. This example is a benchmark from scientific literature, which we used to verify the equations in this fluid flow interface.

📷 An air bubble penetrates the phase boundary between water and oil and entrains a small amount of water in its wake. The entrained water droplet forms a tail behind the rising bubble.

This problem is interesting in a microfluidic system, since this mechanism can be used to transport small droplets of water into a layer of oil. The water droplets can, for example, be used to extract water soluble species from the oil into the water droplets, while keeping the hydrophobic species in the oil, to perform separation in a very controlled way. If the size of the water droplets is small enough, coalescence of the droplets in the oil phase may be avoided, thus creating droplets with a specific content and weight.

The same model can also be used to calculate the size distribution and coalescence kinetics, which can in turn be used in a dispersed multiphase flow model of an air-water-oil mixture. Emulsions can be used to create powders and structured mixtures.

The Physics and Model Behind the Three-Phase Flow, Phase Field Interface

The schematic below shows the three immiscible phases. The model is based on a free energy formulation of the system using three different phase field variables, with one for each phase (A, B, and C). The phase boundary is determined by the isosurface of a phase field variable for the value of 0.5, which corresponds to the pink and gray isosurfaces in the image above. The sum of all phase field variables in each point in space has to be equal to 1. The phase field variables are thus measures of the content of each phase in every point in space.

📷 A schematic drawing of the three-phase system, visualized in a projection plane perpendicular to the container walls.

The free energy equation is a function of the phase field variables and the surface tension for each pair of possible boundary interfaces, i.e., AB, AC, and BC. Each of the phase field functions is then used in the conservation equations for each field, which include the minimization of the free energy of the system. The formulated equations are the so-called Cahn-Hilliard equations.

Note also that this formulation accurately treats the triple point between the three phases, which is the point between the blue, pink, and white colored regions for phases A, B, and C shown above. This allows for the simulation of partial and absolute wetting between the three phases.

The interaction with the walls is determined by the contact angles in figure 2, θi, which are set to fixed values. The contact angles are used to express the boundary conditions for each of the phase field variables at the walls. Each angle is computed as the angle between an isosurface of the phase field variable at the value of 0.5, using the projection on a plane perpendicular to the walls, as shown above.

The surface tension forces in the system are also introduced in the equations for the conservation of momentum (Navier-Stokes equations) as sources of momentum. The density and viscosity, at each point in space in the equations for conservation of momentum and mass, are computed from the phase field variables, switching from the values of one fluid to another. Each phase gets the density and viscosity of the pure phase, which smoothly but rapidly changes across the phase boundary at the phase field value of 0.5.

The formulation described above is the one used in the phase field method, which is considered one of the most accurate ways of describing multiphase flow in continuum models.

An Intuitive User Interface

The user interface in the Three-Phase Flow, Phase Field interface in the CFD and Microfluidics modules is a so-called multiphysics interface. This means that, as a user, you have control of both the Cahn-Hilliard equations for the phase fields and the fluid flow equations. Although the settings are available in predefined formulations, an experienced user may also easily extend the equations to include other phenomena; for example, electric fields for studying electrocoalescence.

The image below shows the physics interfaces to the left, in the model tree, which are defined by the Three-Phase Flow, Phase Field interface. The included physics interfaces are the Laminar Flow and Ternary Phase Field interfaces. In addition, the Multiphysics node couples these two physics interfaces in its child node, the Three Phase Flow, Phase Field coupling node.

In the Ternary Phase Field interface, the Mixture node settings contain the input fields for surface tension, as shown below. In addition, the convection term is displayed, showing that the velocity field is obtained from the coupling formulated in the Three Phase Flow, Phase Field coupling node.

📷 The Mixture node contains the settings for the Cahn-Hilliard equations, which are the surface tension for the mixture and the coupling velocity field. The Equation section shows the domain equations.

The interaction with the container wall is defined by the settings for the Wetted Wall node, shown in the image below. Here, we find the input fields for the contact angles and a description of the notations used for these angles.

📷 Settings for the Wetted Wall boundary condition, which sets the contact angles for the different phase boundaries with the walls of the container.

The settings for the Three Phase Flow, Phase Field coupling node are shown below. Here, we can see that the coupled physics interfaces are the Laminar Flow and Ternary Phase Field interfaces. For an advanced user, the coupling node gives the possibility to couple the Ternary Phase Field interface to different fluid flow interfaces, which may be defined in different ways or in different domains.

📷 The settings for the Three Phase Flow, Phase Field coupling node.

Possible Future Extensions to the Functionality

The first version of the Three-Phase Flow, Phase Field interface is formulated for laminar flow problems. A natural extension is to also formulate this model for turbulent flow. We are planning to offer this capability in a future release of COMSOL Multiphysics. Another natural addition is to include solid particles in the flow. This can, in fact, already be done using the Particle Tracing for Fluid Flow interface. We also plan to provide related Application Library examples in future software versions.

Further Reading

Categories

📷

Suggested Posts

📷

The Beauty of Vortex Streets

📷

Designing District Heating Networks Using Topology Optimization

📷

8 Uses of COMSOL Multiphysics® in the Biomedical Industry

Yes! Thanks!

Hi Joseph!

No, not actually. i want to simulate a channel with 3 inlet (They are all liquid). i triied to use Two phase levelset ... but i could not. as in the level set it asks me like ꬾ = 1 or 0 for Fluid 1 and Fluid 2. But there is not any for fluid 3. I dont know how to introduce fluid 3. Can you help me with this please, its many days that i could not solve it .. :( Joseph Ackermann

Okay, thank you:)

I tried MATLAB, COMSOL and writing my own script, in my opinion, if develop toward theory side, that is when computing power is less required, i recommend MATLAB, if going application side, I recommend building your own script, phase-field simulation is very computing demanding, COMSOL apply FEM to solve the equation, this limits the available grid points hence the simulation domain size.

Nothing referring to experiment has been scheduled. There is no such device or condition. What's more, which method to use in effect depends on whether other have researched and published. As a graduate I'm not powerful enough to independently apply some method on my project, what I can do is take a page from other papers.

Also my tutor is not major in this scholarly orientation and even, I guess, a paper related to it has not been read, because he is not clear what I'm talking about while group meeting. The phase field method is the priority just on account of the former project which I used phase field to establish.

He is afraid of adventure, a new method means a vague future. But I have read a lot of paper, all the paper simulate bubble/water kinetics referring to concentration variation use the VOF method. Here is my doubt, no one paper using method field not means this method is unfeasible, probably I haven't find it. Consequently, I ask for help here to some expert.

We are not in the same environment, not in the same condition, something may be incredible for you but normal for me.

Anyway thank you for replying my post, but I hope a judge based on academic theory.

You may look into the Eq. 23 of the paper authored by Abinandanan and Haider (Philos Mag A 81 (2001) 2457) for an extended Cahn-Hilliard model, if you are still looking for an answer. From your query it seems the model you are dealing with is an Allen-Cahn type phase field model. In such a case, if you do not have any higher order terms, then by simply ignoring the third term in Eq. 23, you should be able calculate the total free energy of the system.

Hello,

There is a stability function in Thermo-Calc that gives the second derivative of G. It is called QF(phase) defined as the determinant of -(d²G/dx²).

I hope this can help.

Regards

Hello,

Q: How long simulation time is sufficient for phase field modeling using COMSOL?

ANS: I used Fortran language and use the other one to represent the graphic.

That was for long time ago. I think the time is near.

We have tried AMR for multiphase flow with phase-field in the past. Usually, refinement is dependent on the phase-field parameter and you can control the number of refinements in a given time scale of simulations. You can refer to the following articles for more information on the implementation of AMR in the Comsol phase field.

1) Yue, P., Zhou, C., Feng, J. J., Ollivier-Gooch, C. F. and Hu, H. H., "Phase-field simulations of interfacial dynamics in viscoelastic fluids using finite elements with adaptive meshing," Journal of Computational Physics, 2006, v. 219, n. 1, pp. 47-67.

2) Zhou, C., Yue, P., Feng, J. J., Ollivier-Gooch, C. F. and Hu, H. H., "3D phase-field simulations of interfacial dynamics in Newtonian and viscoelastic fluids," Journal of Computational Physics, 2010, v. 229, n. 2, pp. 498-511.

3) Yue, P., Feng, J. J., Liu, C. and Shen, J., "A diffuse-interface method for simulating two-phase flows of complex fluids," Journal of Fluid Mechanics, 2004, v. 515, pp. 293-317.

4) Yue, P., Zhou, C. and Feng, J. J., "Spontaneous shrinkage of drops and mass conservation in phase-field simulations," Journal of Computational Physics, 2007, v. 223, n. 1, pp. 1-9.

5) Yue, P., Zhou, C. and Feng, J. J., "Sharp-interface limit of the Cahn–Hilliard model for moving contact lines," Journal of Fluid Mechanics, 2010, v. 645, pp. 279-294.

6) https://www.emerald.com/insight/content/doi/10.1108/HFF-01-2020-0001/full/html

Aynur Atalay Try setting chi higher, e.g. more than 10.

Nikolas Provatas' book on phase field methods is a good starting point:

H, Fan Peng, Were you able to solve the divergence and mesh distortion problem in ABAQUS?

Hi,

here comes a recent preprint of mine, where phase-field helped (at least me) to understand and perhaps even further develop concepts of analytical philosophy.

The article is named "A Phase-Field perspective on Mereotopology" and it is available here:

Dear Viswanathan Ramamoorthy, considering the sum constraint, that you also show in the image, the gradient energies in the ternary alloy are not independent, i.e., this can be just written such for A, B and C but only two of these three gradient terms are independent. If you check the corresponding evolution equations for concentration fields, this must become clear: dC_A/dt + dC_B/dt + dC_C/dt = 0.

The source of gradient energies is the spatially asymmetric interactions between the atoms/solutes in a given gradient. If you like to understand the true physical origin of it, I suggest you look into these derivations

Dear Vishesh,

As the phase-field method developed very gradually, each part of it was improved in some steps. First, the so-called gradient-term was recognized to be important (van der Waals, Hillert, Cahn &Hilliard, Langer, etc.), then temperature (for studying solidification), concentration field (for alloys), and elasticity were gradually included and also some other energy densities like dislocation/plasticity, electromagnetic etc. Each with some varieties and each through important contributions indeed! There are a couple of key review papers like these two

that can help. And I suggest you read the depth of it in early works of those names mentioned above. In one recent work I tried a derivation from scratch that might interest you: .

The fact is that phase-field type descriptions are still developing and very likely will be that way for a while: This is both a blessing and curse ---a blessing because it's capable of addressing complex, emerging cases, and a curse because its descriptions and implementations must update over and over.

Luigi Candido Thank you for your answer. I have never used Simulink. I will have a look.

Dear Mehdi Shahedi Asl, Mehdi Akbarifar

You can find it in coefficient form PDE module. Please note that, you should define the units of your parameters.

You can, also, use the pdf form manuals which provided by software.

Regards,

Farhad

I suggest you that take a look at following link.

Dear Dr. Chenxi Wang, there exists a formulation of the ferroelectric phase-field model which considers the piezoelectric effect. You may be interested in the work of Schrade et al. (see https://doi.org/10.1007/s00419-013-0754-5 and references therein). They consider a coupling between the strain and the electric field in the free energy of the material.

I cannot comment about the applicability of these methods to.simulate the solidification of aluminum castings. However, I would.like to caution you that pores are extrinsic defects, caused by entrainment damage. Without any entrainment damage, pores, whether shrinkage or hydrogen, cannot nucleate even heterogenously. Therefore the entrainment damage needs to be known for simulation. This is extremely complex and damages are additives, i.e., ingots come usually with extreme levels of damage and there is more damage in the production system as well. Therefore it is quite a challenge to simulate pore formation, except in badly damaged castings, where you can assume that pores will form when a certain level of hydrogen or negative pressure level is reached.

With Mathlab ,we have :

Dear Nils, I do not know if this helps.. I am trying to organize a special issue of the open journal ( Metals) . Originally the idea was on ( elementary) Calphad tools for solidification.. Now I am moving to the idea of using the special issue as a container for something in between remote learning and remote collaboration for students ( of any age!) that will not have the possibility to cross easily borders - like most of them have been doing in last years. Something in between a summer school and an internship in a foreign institution/ firm. If you or your collaborators would have the time, we could publish your work there also using the remote learning as tool to refine the usability of the model. Kindly let me know if you are of the idea, thanks FM

Jiří Kroc many thanks.

Amir Reza Ansari Dezfoli Very grateful.

I dont know what about? but it is just a suggestion make mean the mid point of color map

I cannot open the attached file. Can you explain in simple mathematical terms the problem you are solving, and the method you have chosen, until the point where you would like advice.

Then I may be able to help. I have used algebraic properties of systems for reduced complexity, and sparse data are one case...

Guangpu Zhu

Hi, I was able to to it. Thanks for your reply. It was really helpful.

The chemical potential of a given species represents the potential or the force that ensures the transfer or movement of that species between two phases. The transfer stops when this potential becomes the same in both phases. This potential depends on the temperature, pressure and composition of the species in the phase considered. The potential of the species is the same in both phases does not imply that the composition of the species is the same in both phases.

To better understand this phenomenon, it must be compared to heat transfer. Indeed, temperature is the heat transfer potential. When two bodies are brought into contact, heat will be transferred from the hot body to the cold body and heat balance will be obtained when the temperatures of the two bodies become equal.

+ two phases

+ computationally less expensive (one transport equation)

+ recommended for larger scale simulations where the interface is not well resolved by the mesh and where the mean position of the interface is sought rather than the fine details.

+ can solve for up to three phases.

+ allows for fluid-structure interaction and phase separation models.

+ includes more physics and it is more accurate as long as the interface is properly resolved by the mesh.

+ computationally more expensive (two additional transport equations)

+ recommended for microfluidic simulations where the surface shape is of primary importance.

Source: https://www.comsol.com/blogs/which-multiphase-flow-interface-should-i-use/

- level set method: capillary filling

- phase field method: Two-Phase Flow with Fluid-Structure Interaction

Dear Vitesh Shah

Its up to the problem you want to solve. In any case, selecting a method with less computational cost but high accuracy is preferred.

But you have to note that in some cases, we have to select multi-phase filed model. Consider a solid phase formation (melt to solid or solid to new solid phase), in this case each nucleation (new solid grain/phase) can have different physical properties. Then WE HAVE to use multi-phase filed model.

Let me give you more specific example;

Assume we have silicon melt and then we want to simulate solidification. During the solidification, each new nucleated grain has different crystallographic orientation and then has different grain growth rate as a function of under cooling temperature. Then during the solidification (phase field modelling) they shows different behavior. Thus we have to use multi phase field model with different physical coefficient in the free energy at phase field model.

Regards,

Hi Diwakar,

I think you already know the answer and just asked to confirm.

Your explanation is very clear.

Regarding third case,injectant fluid- liquid( 120 K, 60 bar) i.e, Tr=0.8 & Pr=1.2 and supercritical chamber ( T = 180 K, P = 55bar)Tr=1.2&Pc1.1,

Since both the injectant and chamber pressure are supercritical, depends on the temperature at the interface region, the surface tension come into play.

For example, if region with Tr < 1 surface tension,and Tr >1no surface tenison.

Hope works of Arnab et al (refer attachment) would be helpful

Regards,

Senthil

thanks so much

Santos H.A.F.A., Silberschmidt V.V. (2015) Finite Element Modelling of 2D Brittle Fracture: The Phase-Field Approach. In: Silberschmidt V., Matveenko V. (eds) Mechanics of Advanced Materials. Engineering Materials. Springer, Cham

The connection between diffusivity and mobility M=D/kT is known as The Einstein- Nern

st equation in the absence of the chemical factor, which may be easily derived from the drift-diffusion equation in the presence of the additional driving force, which is a gradient of the applied external field at the steady state considerations,only!!!.

Thanks a lot for your kind references.

Deewakar

The classical paper on the subject is a paper in PRA by Gunduz Caginalp.

I do not know the name, but it is easy to find... oh found it

Entropy: A concept that is not a physical quantity

You can take dimensional data as data using natural units https://en.wikipedia.org/wiki/Natural_units

Then dimensionless number are all physically unit data

Here is an intro example:

Hi Amir,

Since your question is mainly about modeling and programming resources, I strongly recommend to use OpenPhase: http://www.openphase.de/ which is an open source software for Serval phase field models. I guess it will help at the beginning with your initiation in phase field modeling and its implementation.

Good luck.

The question is fundamentally superficial. For some researchers it's just a different terminology for the same thing. For the others the differences are not the name but the derivation of their theories (thermodynamic approach, variational approach), constitutive laws (I mean for someone who understands, the local damage dissipation function w(alpha)), numerical treatment (how to numerically implement irreversibility for instance), etc...

If you are interested you may read the first section of my thesis:

Well they do work at different scales. You use phase field to model the response of a system at the macro or micro scale similarly to continuum mechanics (micron up to cm scale or even m scale).

While MD is discrete mechanics modeling individual atoms etc. So you are down to Angstrom or nanoscale.

Usually you calculate material properties with MD and feed them into your macroscopic phase field model.

MD is computationally demanding while phase field is over simplified technique.

It isn't right to compare which one is better; as they are complementary to each other. You could use both of them in order to develop a more microstructure-informed model.

A) First part of the answer

When the Navier-Stokes/Cahn-Hilliard model is used, the Korteweg tensor is often expressed in its equivalent potential form -C grad(mu) in Navier-Stokes equation where C is the composition and mu the chemical potential. You can see the following reference p 153 (relationship 2.9)

In equation (2.10) both terms of the Korteweg tensor appear.

To prove the equivalence, it is easier to consider the notations with the Einstein summation convention and the kronecker symbol. To guide you to prove it, you can see few details in the book http://eu.wiley.com/WileyCDA/WileyTitle/productCd-1118971337.html

calculus (9.10) p 294.

B) Second part of the answer

A quick proof of the Korteweg tensor is derived p 143-144 in

C) Last part of the answer

Another method is possible in order to derive the Korteweg tensor.

The demonstration starts from the differential of internal energy du with an important additional term : dot product of d(grad rho) and its conjugate vector (chosen K.(grad rho)) where K=Kappa is the capillary coefficient.

Next, use the conservation equations du/dt, ds/dt and du/dt (where s is the entropy) and manipulate the term K.grad(rho).d(grad rho)/dt. Finally express the entropy source and you will derive the relationships of the stress tensor T and the heat flux q with the term K.grad(rho). For T, the result is the Korteweg tensor.

Please evaluate some manuscripts from my library.

Using classical XFEM for fracture mechanics (i.e. existing crack propagation) is perfectly usable in thermo-elasticity: both displacement and temperature fields are then enriched and mechanical strain energy used in the computation of the J-integral must be updated accordingly. You can find all this in the following reference:

Duflot, Marc. "The extended finite element method in thermoelastic fracture mechanics." International Journal for Numerical Methods in Engineering 74.5 (2008): 827-847.

As mentioned by Yiming, a damage model has to be considered to deal with crack initiation. Several regularization techniques have been proposed over the years and methods as phase-field and Thick Level Set can deal with the transition from damage to fracture.

@Deewakar  I change it by the exact doc that I want upload. Good luck.

Many thanks. I'll have a look at the references.

Both the second and the first order phase transitions are defined and treated in the framework of thermodynamics. The major difference between the two types of transitions is the existence of the thermodynamic barrier (>> kT) in the case of first order transitions. Sufficiently large stochastic fluctuations are required for such transitions to occur. Stochastic fluctuations can be also treated thermodynamically through the fluctuation-dissipation theorem. Thermodynamic description of the first order transitions is applicable  for the cases of both the sharp and diffuse interfaces between the ambient phase and a new phase nucleus. For example, in the theory of crystallization the diffuse interface approach is more accurate than the sharp interface approximation, which is valid only when the interface thickness is negligible compared with the nucleus size. Under irradiation conditions, two kinds of "forces" are continuously competing with each other. The first one is the external irradiation, which drives the system away from the equilibrium. The second is the internal thermodynamic force moving the system towards the equilibrium. This force can be described thermodynamically through, for example, the chemical potential. In this description both the diffuse and the sharp interface methodologies can be employed. The major problem with the phase-field in the present state is the correct formulation of the thermodynamic "force" through the corresponding thermodynamic potential.

I will try to answer this question in a very non-scientific manner.

Imagine a spiral whisk (see enclosed picture). If you keep the spiral part perpendicularly to your eyes and you look through, ignoring the 3D aspect of the object, you will see a modulated sine signal (a 2D projection), lets call it x(t).

The Hilbert transform shifts the 2D projected signal x(t) by 90 degrees in phase to create a signal y(t) = hilbert[x(t)] so that if you combine the original front projection x(t) and its Hilbert transform y(t) you got something we call the analytic signal z(t) = x(t) + j y(t). The analytical signal is, in our case, exactly the spiral whisk.

In terms of mathematics, the front projection of the whisk x(t) is the real part, the top projection y(t) would be the imaginary part, and the whole 3D spiral whisk is the analytic signal z(t), so z(t) = x(t) + j y(t).

Now, obviously, when you take the analytic signal (the spiral whisk), it is easy to extract the envelope part (the absolute value) and the carrier (phase, that can be modulated).