A comprehensive description of the phase change process in general terms started with the early work of Lamé and Clapeyron (1831), where solidification due to heat conduction was the driving catalyst for the phase change process. Based on their findings, a new class of models emerged for the Lamé-Clapeyron-Stefan problem, and results related to that problem are discussed in Beckermann, Diepers, Steinbach, Karma, and Tong (1999) and Rubinšte ǐn (2000). These models basically consider a general approach based on the heat equation with free boundaries that takes into account some thermodynamic aspects of heat conduction for the initial and boundary conditions, and mass flux via the Navier-Stokes equations (Gupta, 2017;Hu & Argyropoulos, 1996). A great deal of mathematical work has been carried out with regard to the problem of solidification and melting and a detailed account of the same with copious literature can be found in Bankoff (1964), Crank (1956), and Rubinšte ǐn (2000). On the physical side Avrami wrote a series of papers concerning the equations underlying the phenomenon of phase change (see Avrami, 1939Avrami, , 1940Avrami, , 1941)). Baldoni and Rajagopal (1997) have also proposed a thermodynamics for solidification.
We should mention closely related papers wherein the same thermodynamic framework is used to develop constitutive relations when polymeric materials are undergoing phase change. Early studies of the same are due to Fasano, Kannan, Mancini, and Rajagopal (2007), Kannan, Rao, and Rajagopal (2002), and Rao andRajagopal (2001, 2002). Some relevant later studies concern phase change in polymers that are light activated with the added property of having memory for certain preferred shapes (see Barot, Rao, & Fig. 1. Illustration of the R2RNIL sub-processes (mold filling, phase change, and demolding) and the regions of the mold roller and time intervals for each sub-process. In mold filling, the stress-free fluid film on the substrate flows into the mold pattern cavities, which is followed by the phase change sub-process in which the substrate is irradiated with UV light to cure the coated film in the pattern. A demolding roller is employed to peel-off the patterned substrate from the mold roller and the substrate is transported downstream.
irradiating it with UV light, which along with the conservation laws results in a model consisting of a well-posed system of partial differential equations. The corresponding initial conditions are determined by the final conditions of the linearized mold filling problem (Gomez-Constante et al., 2019, 2020). The boundary conditions are determined by the interaction of the mixture with both the tension of the web and the UV light source.
The organization of the paper is as follows. In Section 2, we derive the constitutive equations of the constituents of the mixture based on the maximization of the rate of entropy production. Section 3 deals with the photo-chemistry of Photo-tunable Molecular Crosslinking (PMC) and the Radiative Transfer Function. In Section 4, we derive the initial and boundary conditions by taking into account the information about the final state of the previous mold filling process. Section 5 provides the general nonlinear system of equations for the phase change process and in Section 6 we obtain a quasi-linearized version of the nonlinear model that enables a numerical approximation that still exhibits the qualitative behavior that we conjecture is adequate for phase change. Section 7 is concerned with numerical simulations of the process depicting the more important qualitative behavior of the model. Conclusions and future directions are provided in Section 8.
We will develop a model that considers the general theory of the mechanics of mixtures (Rajagopal & Tao, 1995) along with the assumption that the rate of dissipation function due to mechanical working on each constituent is maximized. We will find the constitutive equations that both constituents of the mixture must have in order to satisfy the conservation laws.
Let us consider a mixture of two constituents: a viscoelastic fluid (or melt) and a viscoelastic solid. We shall denote with subscripts 𝑚 and 𝑠 the melt and solid constituents, respectively. Let 𝐗 𝑖 (𝑖 = 𝑚, 𝑠) be a material point belonging to constituent 𝑖 in the reference state. The motion of each constituent is defined through
Assuming the motion to be adequately smooth, the deformation gradients (𝑭 𝑖 ) and the left and right Cauchy-Green tensors (𝑩 𝑖 and 𝑪 𝑖 ) are given by
Assuming there is no relative motion between the constituents of the mixture, i.e.,
the velocity gradient of the constituents is given by
In addition, if we assume and that the temperature of the mixture (𝜃) is the same for each constituent, i.e.,
and both constituents coexist at the same material points, then by additivity of mass, the density of the mixture (𝜌) satisfies
Now, the conservation of mass for each constituent of the mixture is given by
where 𝑚 𝑖 is the mass production rate of each constituent 𝑖. Since the mass fraction of melt changing phase into solid must be equal to the mass fraction of solid created, they must satisfy
Substituting Eq. ( 6) into Eq. ( 7) we obtain the conservation of mass of the mixture as
The angular momentum supply is asumed to be zero and the Cauchy stress associated with the constituents are assumed to be symmetric. Using a similar reasoning and assuming that each constituent satisfies the conservation of linear momentum, we get that the conservation of linear momentum for each constituent and the mixture to be given by
where 𝑻 𝑖 and 𝒃 𝑖 , respectively, are the Cauchy stress tensor and body force of constituent 𝑖, and 𝒎 𝑖 is the momentum generated by mass production of constituent 𝑖 which satisfies
The Cauchy stress tensor (𝑻 ) and the body force (𝒃) acting on the mixture are given by
The conservation of energy for each constituent and the mixture are given by
where the heat flux (𝒒), internal energy (𝜌𝜖) and radiant heating of the mixture (𝜌𝑟) are defined as
The energy supply (𝜖 𝑠 𝑖 ) of both constituents satisfy
Finally, by defining the Helmholtz potential function (𝜓) and the entropy (𝑠) of the mixture as
we find the reduced energy-dissipation equation of the constituents and the mixture as
By defining the mass fraction of the melt as
where 0 ≤ 𝛼(𝒙, 𝑡) ≤ 1, we can now see that the body force, internal energy, radiant heating, Helmholtz potential and entropy of the mixture are defined as a convex combination of the mass fraction of both the melt and the solid. In addition, we have
Substituting Eq. ( 25) into Eq. ( 23) and comparing it with the sum of the individual constituents defined by Eq. ( 22), we have
From Eqs. ( 7) and ( 26), we have
which basically states that the change in the mass fraction of the melt is proportional to its mass production rate per unit volume. By using the conservation of mass and the definition of Helmholtz potential of each constituent, we obtain
Substituting Eq. ( 28) into Eq. ( 22) yields
We next define
where 𝜁 𝑖 , 𝜁 𝑝 𝑖 and 𝜁 𝑐 𝑖 are the rate of entropy production due to mechanical working, phase change and heat conduction, respectively. We will assume that they are all non-negative scalar functions. In order to find the appropriate constitutive equations for our mixture, we will use a thermomechanical approach consisting of the maximization of the rate of entropy production of mechanical working of the mixture defined as 𝜁 = 𝜁 𝑚 + 𝜁 𝑠 (33) along with the restriction of isochoric motion enforced on the natural configurations explained below.
For what comes next, we assume that both the melt and solid constituents of the mixture have natural configurations 𝜅 𝑝 𝑚 (𝑡) and 𝜅 𝑝 𝑠 (𝑡) , respectively (Fig. 2). Using these natural configurations, we can now define the mappings from their respective reference configurations (for the melt and the solid) as
(34)
where the tensors associated with the natural configurations for the melt and the solid are 𝑭 𝜅 𝑝 𝑚 (𝑡) and 𝑭 𝜅 𝑝 𝑠 (𝑡) respectively. Now, we proceed to define the velocity gradient and the mappings associated with both natural configurations as
which then allows us to define their symmetric parts as
For the sake of simplicity, we assume that the Helmholtz potential 𝜓 𝑚 and the rate of entropy production 𝜁 𝑚 of the melt are given by
where 𝜌 0 𝑖 is the initial density of the constituent and 𝐺 𝑖 , 𝜇 𝑖 are material coefficients related to each constituent and 𝜃 0 is the initial temperature of the mixture.
Maximizing the rate of entropy production (Eq. ( 33)) along with the restrictions of both natural configurations associated with each of the constituents being isochoric, we have tr 𝑫 𝜅 𝑝 𝑖 (𝑡) = 0 (43)
Further, the Cauchy stress tensor of the melt 𝑻 𝑚 is given by
and the Cauchy stress tensor of the solid 𝑻 𝑠 is given by
where ∇ 𝑩 is the upper-convected time derivative defined as
Rearranging terms and considering that ∇ 𝑩 𝜅 𝑟 = 𝟎, the constitutive equations of the melt and solid are given by
and
Note that the constitutive equations describe the behavior of the melt and solid constituents of the mixture independently. However, this formulation does not explain how the densities of the constituents evolve in time due to UV light irradiation; to describe this we need the governing equations for the photochemistry of the phase change of the mixture which we describe below.
Photochemical reactions are a special class of chemical reactions that are activated by irradiation of light of certain energy determined by their frequency. Photons of certain frequency (in the ultraviolet electromagnetic spectrum in our case) are the ones that trigger the chemical reaction by adding energy to the molecules and interacting with the valence electrons of the molecules composing the polymeric chains used for this manufacturing process.
Generally speaking, there are two main types of photo-chemical reactions: Photo-Induced Network Rearrangement (PNR) are chemical reactions in which the initial molecular chain is broken by the irradiation of UV light, producing two smaller molecules one of which is the one that possesses the desired mechanical properties; and Photo-tunable Molecular Crosslinking (PMC) which is an internal reconfiguration of certain parts of the chain of the polymer that are triggered by the irradiation of UV light and produces the stiffening of these long chains, macroscopically producing the desired mechanical properties. PNR reactions need additional stoichiometric balance equations along with some extra terms that will appear in the reduced energy-dissipation equation (Eq. ( 22)) making this approach more complicated, whereas for PMC reactions all we need is an experimentally determined equation relating the amount of polymeric fluid being converted from one phase into the other in terms of time. Both PNR and PMC reactions are governed by the same physical principle that governs the interaction between matter and electromagnetic radiations.
J.P. Gomez-Constante et al.
Let us suppose that we have an electromagnetic radiation field emitted by a source that passes through a medium. The amount of energy in the form of radiation that is absorbed by the medium, transmitted through the medium, and scattered in different directions across the medium is governed by the equation (Chandrasekhar, 2013)
where 𝑐 is the speed of light, 𝐼 is the intensity of the light through the medium, 𝜎 𝑠 is the absorbance due to the scattering, 𝑎 𝑚 is the absorption coefficient of the melt, 𝑐 𝑚 is the molar concentration per volume of the melt, 𝑎 𝑠 is the absorption coefficient of the solid, 𝑐 𝑠 is the molar concentration per volume of the solid, 𝑗 𝜈 is the radiation emitted by the medium and 𝐼 𝜈 is the scattering kernel that takes into account the non-homogeneous nature of the medium. Since we are not dealing with radioactive materials and the scale of the process is very small, it is reasonable to neglect scattering and emission, which reduces the equation to:
where r is the unit radial vector along the direction in which light propagates inside the medium. Notice that if we neglect the time derivative in the previous equation then we recover the well known Beer-Lambert Equation.
Conservation laws alone cannot describe the response of bodies, one needs constitutive specifications, similarly the radiative transfer equation alone is not enough to describe photo-chemical reactions. Additional equations describing the particular behavior of the molecular chains forming the materials in the mixture is needed. The process for obtaining such equations is out of the scope of our work, so we will just use a rather simple specification. We will assume that the variation in time of the molar concentration of the melt is given by the equation
where 𝑁 𝐴 = 6.022 × 10 23 mol -1 is the Avogadro's constant, ℎ = 6.626 × 10 -34 J ⋅ s is the Planck's constant, 𝜈 is the frequency of the photon and 𝑐 0 is the concentration of the mixture at the beginning of the process. In this particular case, we have assumed that the photo-chemical reaction is bi-molecular which is why the exponent of the molar concentration is equal to 2 (Long, Scott, Qi, Bowman, & Dunn, 2009). In addition, the molar concentration per volume of a chemical component 𝑖 is related to density via
where 𝜌 𝑖 is the density of constituent 𝑖 and 𝑀 𝑖 is its corresponding molecular weight. With this relation between the density and molar concentration per unit volume for each constituent, we can now express how the densities of both constituents will evolve in time in terms of the thermomechanical and the photochemical behavior.
Let us assume a system consisting of three parallel rigid rollers: a mold roller of outer radius 𝑅 𝑜 rotating at constant angular speed 𝜔, a nip roller producing the filling of the cavities imprinted in the mold roller and an auxiliary demolding roller. The separation between the secondary rollers produce a fixed wrapping angle which for the sake of simplicity we will set as 𝜑 = 𝜋∕2 thus determining the arc which is the zone where phase change will take place (Fig. 3).
In addition, we will assume that mold filling has successfully been accomplished in the previous sub process meaning that the mold cavity has been filled with fluid up to a given minimum accepted tolerance.
Since the angular speed is assumed to be constant, the available time for changing phase from a viscoelastic fluid to a viscoelastic solid denoted as 𝑇 2 -𝑇 1 is given by
For phase change, the boundary condition is given in terms of the normal reaction that balances the tension of the web that acts as a traction force on the substrate. To elaborate, let us consider a small portion of the substrate 𝜑 (Fig. 4).
To simplify things a little bit, it suffices to assume that the tension of the web ⃗ 𝑡 𝑤 is uniform across the substrate thickness and in the tangential direction. This tension along with the curvature of the roller produces a reaction force in the normal (radial) direction which keeps the fluid inside the mold cavity from being expelled by both the non inertial forces and the energy stored from the previous mold filling sub process. Therefore, the boundary condition at the substrate is
where 𝑻 𝑠𝑢𝑏𝑠 , 𝑻 are the Cauchy stress tensors in the substrate and the melt respectively and n is the unit outward pointing normal in the substrate direction.
J.P. Gomez-Constante et al.
Since the mold patterns and the fluid film are in the nano-scale, most of the energy irradiated from the UV light source will not be absorbed by the fluid. In fact, we expect that the vast majority of that energy will be absorbed by the mold roller and reflected back to the surroundings making this a possibly temperature-dependent process. We identified two main interactions between the fluid film and its surroundings: heat conduction between the fluid and the mold and heat convection between the fluid and the air trapped inside the mold cavity (Fig. 5).
Assuming the manufacturing process runs at constant parameters, the interaction between the fluid and the mold can be modeled as a constant heat flux ⃗ 𝑞 𝑚𝑜𝑙𝑑 that flows from the mold roller into the fluid, increasing the temperature of the fluid
where 𝑘 is the conductivity of the fluid. For the interaction in between the fluid and the air we can assume that convection of the fluid is given by
where ℎ 𝑐 is the heat transfer coefficient and 𝐴 is the area of the free surface. The latter applies to the interaction between fluid and substrate because of the lumped heat capacity assumption.
J.P. Gomez-Constante et al.
Similar to what happens with rays of sunlight irradiating our planet at human-scale structures, the nano-scale of the features makes it possible to assume that UV light rays irradiating each mold feature are parallel and that only their direction (which is now uniform) will change with time.
For the sake of simplicity, we will assume that the source of UV light is located at a point fixed in space, say 𝑄, such that it irradiates a total angle of 𝜋∕2 rad (Fig. 6). This has been done with the purpose of avoiding multiple light sources and all the modeling complications arising from such an assumption. Let us think of a fixed material point in the mixture denoted by 𝑃 . As it travels across the arc of length 𝜋 2 𝑅 0 , the light rays form an angle 𝜙 𝑙 between the normal to the surface at point 𝑃 and the straight line joining point 𝑃 and the position of the light source 𝑄. At position 𝜃 = 𝜋 4 the triangle formed by points 𝑃 , 𝑄 and 𝑅 degenerates into a straight line to then reverse its direction. With all these considerations, the angle formed between the normal direction at point 𝑃 and the UV-light ray is governed by the equation
Now, let us consider what happens with the mixture inside one feature. First, the Law of Refraction (also known as Snell's Law) states that the angle of refraction depends on the medium in which light travels, and that between two adjacent media with flat boundary (Fig. 7) it is governed by equation
where 𝜙 𝑙 is the angle at which light travels through air, 𝜙 𝑚 is the angle at which light travels through the mixture, 𝑛 𝑎 is the refraction index for air and 𝑛 𝑚 is the refraction index for the mixture. In order to simplify the analysis, we will not take into account the light that will be reflected in the mixture/mold-roller interface which will provide some additional energy for the phase change process.
With all the previous considerations, we can finally determine the direction r (in the radiative transfer function (Eq. ( 54))) at which light travels inside the mixture as
where ê′ 𝜃 is the tangential direction along the motion (horizontal), ê′ 𝑟 is the radial direction (vertical) and sin 𝜙 𝑚 and cos 𝜙 𝑚 are given by the equations
As for the boundary conditions, we will assume that the intensity of the light that reaches the outer surface of the mixture remains constant throughout the whole phase change process (which is not true since the curvature of the roller and the absorption of air will affect the intensity that reaches the outer boundary at each position across the arc), whereas the in the interface between the mixture and the mold roller we assume a constant light intensity flux which depends on the light intensity at that boundary, both expressed as:
respectively. These boundary conditions along with the symmetric assumptions on the left and right boundaries of the feature makes the problem well-posed in terms of number of equations and unknowns.
From all the previous analysis and simplifications, we can now state that the governing equations that define phase change of the mixture:
along with the constitutive equations and experimentally determined photochemical behavior defined by
The initial conditions for 𝒗, 𝑻 , 𝜃, 𝐼 and 𝜌 𝑖 are given by 𝒗(𝒙, 𝑇 1 ) = 𝒗 0 (𝒙, 𝑇 1 ) (80)
𝜃(𝒙, 𝑇 1 ) = 𝜃 0 (𝒙, 𝑇 1 ) (82)
𝜌 𝑠 (𝒙, 𝑇 1 ) = 0 (85)
where 𝒗 0 , 𝑻 0 , 𝜃 0 and 𝜌 0 𝑚 are the final conditions of the previous mold filling process and the density of the solid part of the mixture is zero since the fluid is completely melted. The boundary conditions are given by 𝒗 = 𝟎 (at the mold) ( 86)
-𝑘(grad 𝜃) = ⃗ 𝑞 mold (at the mold) ( 88)
All the previous boundary conditions have a periodic condition implemented on the sides. .
The governing equations for phase change as posed in Section 5 are nonlinear and hard to solve. To simplify the problem and in an effort to obtain meaningful solutions, as the first step we will consider a simplified version of the general problem without temperature changes (i.e., 𝜃 = 𝜃 0 ) that can exhibit the basic viscoelastic behavior that we conjecture as appropriate for the phase change process. We will assume that the displacement gradient and its time rate to be small, that is
that the left Cauchy stress tensors of both natural configurations of the melt and the solid portion of the mixture satisfy
that the phase change starts at time 𝑡 = 𝑇 1
and that the density of both the melt and solid parts satisfy
where 𝜌 0 is given and fixed.
With the previous simplifying assumptions the governing equations reduce to Let us further assume that the density of each constituent depends only on time (i.e. 𝜌 𝑖 = 𝜌 𝑖 (𝑡)), then we have that the evolution of the system is given by
where the differential equations describing the evolution of the Cauchy stress tensor can now be solved explicitly and written in the form
Furthermore, we will also assume that the process is bi-dimensional in the sense that any cross section perpendicular to the axis of the mold will have the same strain/stress distribution. By the use of a cylindrical coordinate system aligned with the mold roller, we clearly see that the conservation of linear momentum and mass can be modeled by the equations
where the second and third terms inside the brackets on the left hand side of Eq. ( 113) are the Coriolis and the centripetal accelerations respectively.
Next having in mind that
let us define the non-dimensional variables
where 𝑊 𝑖 is the Weissenberg number, 𝐷𝑒 1 and 𝐷𝑒 2 are the Deborah numbers of the melt and the solid respectively, 𝑅𝑒 is the Reynolds number of the mixture, 𝐾 is the ratio between solid and fluid shear, 𝑃 1 and 𝑃 2 are nondimensional numbers coming from the photo-chemistry of the mixture and 𝑣 𝑠 (𝑡) is given by
where the displacement function of the substrate 1 is defined as
With these definitions and taking into account once more the nano-scale of the features we can show that the differential operators in cylindrical coordinates are approximately the same as their Cartesian counterparts, and that the componentwise conservation of linear momentum and mass are given by
1 for more details refer to Gomez-Constante et al. ( 2019) 𝜕 ū𝜃 𝜕 θ -
where the nondimensional angles φ𝑙 and φ𝑟 are given by
(125)
the non-dimensional Laplacian operator ( Δ) operating on a nondimensional function v is defined as
It is worth mentioning that since the last two equations defining the photo-chemistry of the problem do not depend on ū𝜃 , ū𝑟 nor p, they can be solved independently of the others and then used in the other equations.
Let 𝛺 = (-1 2 , 1 2 ) × (-1, 0) ⊂ R 2 be the spatial domain, and 𝐼 𝑡 = (0, 1) ⊂ R the time interval in which the available time for phase change takes place (Fig. 8). Set 𝛺 𝑡 = 𝛺 × 𝐼 𝑡 as the space-time domain. The process that changes the viscoelastic fluid into a viscoelastic solid is then given by the quasi-linear system of partial differential equations
with initial conditions given by:
periodic boundary conditions on the sides of the spatial domain, and with upper no-slip boundary conditions on the displacements.
For the lower boundary conditions on displacements we will split this into two cases, namely the traction boundary conditions given by
where 𝑛 web is the reaction force due to the tension of the web in the normal direction to the substrate and 𝐴 web is the area of contact between the substrate and the mixture or the non-slip boundary conditions given by
For numerical simulations we use what is know in the literature as the Finite Difference Marker and Cell (MAC) Scheme for free surfaces (Harlow & Welch, 1965). This method consists in discretizing the differential and integral operators in order to generate a mesh on a rectangular domain (Fig. 9) in a way that the discretized equations for displacement and pressure satisfy what is called the inf-sup condition, which is a necessary and sufficient condition for existence and uniqueness of solutions to the system of PDEs defining the mechanical behavior of the problem (Boffi, Brezzi, Fortin, et al., 2013). In addition the MAC scheme enables us to have boundary conditions only for displacement are the natural boundary conditions that arise from the physics of the problem.
In order to simplify things even more, we separated the system of with regard to the mechanical and photo-chemical behavior, that is, we will solve the photochemical reaction (Eqs. ( 131) and ( 132)) to then use the mass fraction of the melt previously obtained to determine the mechanical response of the mixture (Eqs. ( 128)-( 130)). By solving the problem this way we avoid dealing with non-linear equations that arise as a consequence of the conservation of linear momentum and mass and isolate the nonlinearities of the photo-chemistry of the problem in a simpler system of PDE. The numerical scheme to find the mass fraction of the melt and the non-dimensional intensity of the light is a straightforward quasilinear finite difference scheme so its discussion is not relevant. However, the numerical scheme to find the non-dimensional displacements and pressure are worth discussing.
Let ℎ = 1∕(𝑁 -1) and 𝜏 = 1∕(𝑚 -1) be the space and time steps of the discrete problem. At the node 𝑖 at time (𝑛 + 2)𝜏 (Fig. 10) the integral and differential operators involving the horizontal displacement ū𝑥 are given by
where the elliptic operator is approximated as
notice that because of our choice of non-dimensional scales we are able to use the same spatial step for both space directions. By a similar process, we can discretize the rest of the differential equations to ultimately see that the photochemistry of phase change J.P. Gomez-Constante et al. model is described by the nonlinear system of equations:
where 𝑀 1 , 𝑀 2 , 𝐾 1 and 𝐾 2 are the sparse matrices defining the radiative transfer function and the response of the material with respect to the intensity of the light, and the linear system of equations
where 𝐴 1 , 𝐴 2 , 𝐴 3 , 𝐴 3 , 𝐵 1 , 𝐵 2 , 𝐶 1 and 𝐶 2 are the sparse matrices defining the finite difference approximations of the integral and differential operators. Notice that the matrices 𝐴 𝑖 depend on the mass fraction of the melt, but since it has already been calculated in the photochemistry equations, the second system of equations is linear in its variables. In addition, to avoid the restriction on the functional space to which the pressure field belongs, (given by the incompressible mixture assumption) a perturbation in the conservation of mass has been added, where 𝜀 is as small as possible to avoid accumulation of numerical errors.
In order to carry out some numerical simulations, a numerical scheme was implemented in MATLAB ® . The restrictions imposed by the isochoric motion assumption of the mixture affected the pressure terms and ultimately the displacement field. For that reason we added a perturbation of the pressure in the conservation of mass to help alleviate that pressure restriction and improve the numerical stability of the simulations. By doing so we were able to decouple the photochemical behavior from the thermomechanical behavior and solve them independently.
To understand how the UV-light intensity and the mass fraction of the melt affect the photochemistry of the problem, we ran some numerical simulations of their behavior independently of the thermomechanics. From the results we concluded that both the UV-light intensity and the mass fraction of the melt change so little across the domain that it is very accurate to assume that the mass fraction of the melt is spatially uniform and that it only evolves in time. This assumption is what enabled us to take the mass fraction of the melt as a function of time only in the thermomechanical behavior, transforming it into a linear integro-differential system of equations.
Fig. 11 depicts the mass fraction of the melt for different UV light sources. It is clear that the higher the intensity of the UV-light source, the higher the amount of melt being converted into solid. However, it is important to note that the higher the UV-light intensity the higher the amount of heat being irradiated to the surroundings which could possibly affect the isothermal assumption or even the behavior of the phase change process by the amount of energy present in form of heat being absorbed by the mold roller and on the surroundings.
In order to understand how the timeline of the phase change process works we need to understand how the mold filling process works. It starts from an undeformed fluid film at rest which is then squeezed against a mold cavity by the relative motion between the mold roller and the nip roller in order to fill the mold cavity with it. We will assume that the material properties and process parameters are such that the mold filling process has been successfully performed. 2 The final displacement and pressure field of the mold filling process are in fact the initial displacement and pressure field in the phase change process as it was mentioned before. Since photochemistry only takes place in the phase change process, both the mass fraction of the melt and the intensity of the light remain constant throughout the mold filling process, meaning that the laws that govern their behavior are identically zero.
For the sake of completeness in our analysis, both traction and no-slip boundary conditions have been individually considered in the model. In a complete nonlinear model, these two boundary conditions are a single nonlinear combination which represents the conservation of Linear Momentum on the boundary. Fig. 12 depicts the initial and final states of mold filling (upper left and right respectively) and the two possible outcomes of the phase change process. In the first scenario (lower left), the substrate (lower black line) is subject to traction boundary conditions only. This traction force on the boundary is at equilibrium with the external radial (vertical in one mold feature) reaction coming from the tension of the substrate web that keeps the mixture inside the mold cavity. Notice that the vertical fibers on the sides push down to about their original positions whereas the vertical fibers at the center portion of the substrate are pulled up the direction of the mold cavity which is consistent with how the vertical fibers are deformed in the mold filling process. In the second scenario (lower right), the substrate boundary is kept fixed at the same position as the final state of mold filling (upper right) and the boundary condition is given in terms of the classic non-slip condition. In this case, we notice that the recoiling happens sideways since the displacement restriction prevents any vertical motion. We also notice that the mesh fiber structure looks more deformed compared to the traction only boundary condition. In the real manufacturing process we expect to have a combination of both scenarios: the recoiling of the substrate boundary condition due to the internal energy Mixture film (blue fiber mesh) and mold/substrate boundaries (solid black lines) on mold filling and phase change processes for one feature. The mixture starts as a melt at rest (upper left), then the lower boundary moves up causing the mold cavity to be filled if the material and process parameters are adequate (upper right) which also determines the initial state of phase change. From this point on, if the traction conditions are enforced then the fluid film recoils back and the lower boundary is deformed (lower left), but if no-slip boundary conditions are enforced, then the recoiling happens sideways (lower right). stored in the mold filling process and the substrate boundary remaining flat due to the higher relative stiffness of the web substrate. In addition, it must be mentioned that even though both mold filling and phase change processes were modeled as non-dimensional problems, the time scale in which phase change takes place is several orders of magnitude greater than the time scale of mold filling. What this effectively means is that the mold filling process happens so fast that it is virtually an instantaneous process with respect to the phase change process, regardless of the web speed.
Fig. 13 shows the Frobenius norm of strain and extra stress as well as pressure distribution across the nondimensional domain of the feature at the end of phase change when considering the traction boundary condition. It is clear that for most of the domain, the strain and extra stress are very low except for the mold roller boundary. That can be explained by the fact that the fibers in contact with the mold roller do not slip maintaining the strain they had at the moment of contact (at the mold filling process). As for pressure, the values clearly show that it is basically back to zero. Fig. 14 shows the Frobenius norm of strain and extra stress as well as pressure distribution across the nondimensional domain of the feature at the end of phase change when considering the displacement boundary condition. In this case, it is clear that the strain is much higher than for the previous boundary conditions, specially around the upper and lower boundaries. In addition, we notice that the pressure values are several orders of magnitude higher that the values compared to the traction boundary conditions.
In this section of our analysis, we will use both boundary conditions previously described and analyzed and compare their respective effects in order to have a more general idea on what happens in the nonlinear case when the boundary condition at the substrate is given by the conservation of Linear Momentum.
Fig. 15 depicts the 𝐿 2 (𝛺) norm of the extra stress, strain, pressure and the mass fraction of the melt as they evolve in nondimensional time at the web speeds of 1 m/min and 10 m/min. In the extra stress we see a sudden jump at the beginning of the phase change process consistent with the instantaneous mold filling process at both web speeds to then relax to lower values for both boundary conditions. Notice that the higher the web speed the higher the initial stress and that there is a little stress recovery at Fig. 15. 𝐿 2 (𝛺) norm of extra stress, strain and pressure, and mass fraction of the melt for both traction (higher value) and displacement (lower value) boundary conditions as they evolve in time at web speeds of 1 m/min (solid blue) and 10 m/min (dashed red). lower web speeds consistent with the solid behavior becoming more dominant as the mixture changes phase. In addition, we notice that even though the traction boundary condition has lower values, its time evolution exhibits the same behavior. In the figure which depicts the strain, we again notice a sudden jump during the mold filling process to two high values corresponding to the different boundary conditions, that then remains about the same for the rest of the process. In this case, we notice that the higher the web speed the less the strain relaxes, consistent with the higher stress values and the reduced amount of time available for relaxation. As for the pressure, we see that its value drops back to zero for traction boundary conditions whereas for the displacement boundary conditions it jumps to a high value and stays approximately constant for the rest of the process, which is consistent with what we expect. Notice that this behavior is approximately the same regardless of the web speed. Finally, the mass fraction of the melt shows that the amount of melt decreases steadily (thus the amount of solid increases at the same rate) for both web speeds being faster and higher the amount of melt being converted into solid for lower web speeds, as expected.
From the model development and analyses of the phase change sub-process, it is clear that the most important factors affecting phase change are the intensity of the UV-light source and web speed. Since our interest is to increase the throughput of the manufacturing process and based on the simulations we clearly see that any increase of the web speed is accompanied by a detriment of the amount of melt being converted into solid. Therefore, in order to increase the web speed we could potentially increase the intensity of the UV-light source, maximize the wrap angle, etc. to compensate the decrease of the amount of melt being converted into solid.
We can also conclude that the mold filling process is an instantaneous process with respect to phase change regardless of the web speed. Because of this, the elastic behavior will be dominant. However, if we can guarantee that the material properties and process parameters are such that the mold cavity will be adequately filled with the melt, then its impact on phase change is not significant.
With respect to numerical simulations, the isochoric motion assumption enforced on the mixture helps decoupling the photochemistry from the thermomechanical behavior making computations much easier. However, this additional assumption comes at the expense of constraining the functional space to which the pressure fields belong. The perturbation in the conservation of mass helps relaxing this constraint but it also inserts a small amount of error in the computations.
The following are potential directions that we will pursue on this topic in the near future.
1. We plan to conduct focused experiments to corroborate the key features of the model as well as to corroborate the model conclusions from the numerical simulations. These will include testing with various patterns and feature sizes and resin materials on a R2R UV-NIL testbed. We will also design and conduct material tests that can be used to obtain the resin material constants needed for optimizing the transport and process conditions for improving pattern quality. Some of the
To understand what could go wrong with mold filling, the interested reader is referred to Gomez-Constante et al. (2020).