We have derived and modified the dynamical model of a geyser induced by gas inflow and regular or irregular spouting dynamics of geysers induced by gas inflow has been reproduced by the model. On the other hand, though we have derived the dynamical model of a geyser induced by boiling, periodic change between the spouting state and the pause state has not been adequately modeled by the model. In this connection, concerning a geyser induced by gas inflow we have proposed the model as described below. Because pressure in the spouting tube decreases obeying to the Bernoulli's theorem when the spouting state begins and water in the spouting tube begins to flow, inflow of groundwater into the spouting tube occurs. When the amount of this inflow reaches a certain amount, the spouting state transforms to the pause state. In this study, by applying this idea to the dynamical model of a geyser induced by boiling, the periodic change between the spouting state and the pause state could be reappeared. As a result, the whole picture of the spouting mechanism of a geyser induced by boiling became clear. This research results would give hints on engineering repair in order to prevent the weakening or the depletion of the geyser. And this study would be also useful for protection of geysers as tourism and environmental resources.
Drying process of polymer solution coated on a flat substrate is very important in various industrial applications. Then we have proposed and modified a model of drying process of polymer solution coated on a flat substrate for flat polymer film fabrication. In the previous study, we added Marangoni effect as pseudo-negative diffusion on an upper gas-liquid interface to the former model. And we saw through numerical simulation of the modified model that solute on upper gas-liquid interface was attracted more strongly to the edge due to Marangoni effect. In this study, we show that this effect is remarkable on upper gas-liquid interface, while the effect extends to sufficiently inside of the polymer solution film.
We have proposed and modified a model of drying process of polymer solution coated on a flat substrate for flat polymer
film fabrication supposing resist coating process in semiconductor engineering process and so on. And we have clarified
dependence of distribution of polymer molecules on a flat substrate on a various parameters based on analysis of many
numerical simulations of the model. Then we applied the model to thickness control of a thin film after drying through
thermal management. But minute thickness control of a thin film after drying was not enough and more minute thickness
control of it was desired. Therefore, in this study, we add evaporative management for more minute thickness control of
a thin film after drying. As a result, thickness control of a thin film after drying in drying process of a polymer solution
coated on a flat substrate can be improved further through adding evaporative operations to thermal operations
artificially and instantaneously depending on solute's distribution during drying.
We apply the former general dynamical model of drying process of polymer solution coated on a flat
substrate for flat polymer film fabrication to concrete detailed subjects. Concretely we apply the model to
effects of a bumpy substrate as an example. We understand that a humpy structure on a substrate does not
affect nearly solute's distribution after drying because effects of diffusion around the hump are sufficiently
effective as far as the hump interfere with diffusion. We also understand that when the beginning time of
special vaporization near a hump is earlier, solute's distribution after drying except for at the edge's region is
thinner a little because leveling including the edge's region by diffusion is more effective.
We modify the former dynamical model of drying process of polymer solution coated on a flat substrate to deal
with drying process of solution having two kinds of solvents. As a result, we see that there is no essential
difference between solute's distribution after drying in case of having two kinds of solvents and that in case of
having only one kind of solvent and when there are over two kinds of solvents, solute's distribution after drying
obeys the mean value of over two kinds of solvents' parameters, that is, it resembles one simulated numerically
based on the mean value in case of having only one kind of solvent.
We extend the former dynamical model of drying process of a resist solution coated on a flat substrate to
three-dimensional one. And through numerical simulation of the extended dynamical model we clarified
characteristic three-dimensional structure of resist's thickness distribution after drying. And we confirmed
characteristic thickness distribution of resist thin film obtained by two-dimensional model through
three-dimensional model again.
We have proposed and modified a model of drying process of polymer solution coated on a flat substrate for flat polymer film fabrication. And for example numerical simulation of the model reproduces a typical thickness profile of the polymer film formed after drying. Then we have clarified dependence of distribution of polymer molecules on a flat substrate on a various parameters based on analysis of numerical simulations. Then we drove nonlinear equations of drying process from the dynamical model and the fruits were reported.
The subject of above studies was limited to solution having one kind of solute though the model could essentially deal with solution having some kinds of solutes. But nowadays discussion of drying process of a solution having some kinds of solutes is needed because drying process of solution having some kinds of solutes appears in many industrial scenes. Polymer blend solution is one instance. And typical resist consists of a few kinds of polymers.
Then we introduced a dynamical model of drying process of polymer blend solution coated on a flat substrate and results of numerical simulations of the dynamical model. But above model was the simplest one.
In this study, we modify above dynamical model of drying process of polymer blend solution adding effects that some parameters change with time as functions of some variables to it. Then we consider essence of drying process of polymer blend solution through comparison between results of numerical simulations of the modified model and those of the former model.
We have proposed and modified a model of drying process of polymer solution coated on a flat substrate for flat
polymer film fabrication and have presented the fruits through Photomask Japan 2002, 2003, 2004, Smart
Materials, Nano-, and Micro-Smart Systems 2006 and so on. And for example numerical simulation of the model
qualitatively reappears a typical thickness profile of the polymer film formed after drying, that is, the profile that
the edge of the film is thicker and just the region next to the edge's bump is thinner. Then we have clarified
dependence of distribution of polymer molecules on a flat substrate on a various parameters based on analysis of
many numerical simulations.
Then we did a few kinds of experiments so as to verify the modified model and reported the results of them
through Photomask Japan 2005 and 2006. We could observe some results supporting the modified model. But we
could not observe a characteristic region of a valley next to the edge's bump of a polymer film after drying. After
some trial of various improved experiments we reached the conclusion that the characteristic region didn't appear
by reason that water which vaporized slower than organic solvent was used as solvent.
Then, in this study, we adopted organic solvent instead of water as solvent for experiments. As a result, that the
characteristic region as mentioned above could be seen and we could verify the model more accurately. In this
paper, we present verification of the model through above improved experiments for verification using organic
solvent.
We have proposed and modified the dynamical model of drying process of polymer solution
coated on a flat substrate for flat polymer film fabrication and have presented the fruits through some
meetings and so on. Though basic equations of the dynamical model have characteristic nonlinearity,
character of the nonlinearity has not been studied enough yet.
In this paper, at first, we derive nonlinear equations from the dynamical model of drying process
of polymer solution. Then we introduce results of numerical simulations of the nonlinear equations
and consider roles of various parameters. Some of them are indirectly concerned in strength of
non-equilibriumity.
Through this study, we approach essential qualities of nonlinearity in non-equilibrium process of
drying process.
We have proposed and modified a model of drying process of polymer solution coated on a flat substrate for flat polymer film fabrication and have presented the fruits through Photomask Japan 2002, 2003, 2004 and so on. And for example numerical simulation of the model qualitatively reappears a typical thickness profile of the polymer film formed after drying, that is, the profile that the edge of the film is thicker and just the region next to the edge's bump is thinner. Then we have clarified dependence of distribution of polymer molecules on a flat substrate on a various parameters based on analysis of many numerical simulations.
Then we done a few kinds of experiments so as to verify the modified model and reported the initial result of them through Photomask Japan 2005. Through the initial result we could observe some results supporting the modified model. But we could not observe a characteristic region of a valley next to the edge's bump of a polymer film after drying because a shape of a solution's film coated on a substrate in the experiment was different from one in resists' coating and drying process or imagined in the modified model.
In this study, we improved above difference between experiment and the model and did experiments for verification again with a shape of a solution's film coated on a substrate coincident with one imagined in the modified model and using molar concentration.
As a result, some were verified more strongly and some need to be examined again. That is, we could confirm like results of last experiment that the smaller average molecular weight of Metoloses was, the larger the gradient of thickness profile of a polymer thin film was. But we could not observe a depression just inside the edge of the thin film also in this improved experiment. We may be able to enumerate the fact that not an organic solution but an aqueous solution was used in the experiment as the cause of non-formation of the depression.
We have proposed and modified a model of drying process of polymer solution coated on a flat substrate for flat polymer film fabrication and have presented the fruits through Photomask Japan 2002, 2003, 2004 and so on. And for example numerical simulation of the model qualitatively reappears a typical thickness profile of the polymer film formed after drying, that is, the profile that the edge of the film is thicker and just the region next to the edge’s bump is thinner. Then we have clarified dependence of distribution of polymer molecules on a flat substrate on a various parameters based on analysis of many numerical simulations. But above fruits are wholly based on theoretical and numerical studies and verification of the modified model by experiment has not sufficiently done yet except for one by the experiments of dependence of polymer molecule’s distribution on vaporization rate.
In this paper, we verify the modified model by a few kinds of experiments. At first, we verify the dependence of polymer molecule’s distribution on intrinsic viscosity by experiment. In the concrete, we verify the characteristics that the smaller intrinsic viscosity is, the lower height of the edge’s bump is, the shallower depth of the valley next to the edge’s bump is and the more flat profile of polymer molecule’s distribution is at central region. Then we verify the dependence of polymer molecule’s distribution on the ratio of molar volume of solute to one of solvent by experiment.
We did many numerical simulation of the modified model of drying process for the flat polymer (resist) film fabrication taking effects of latent heat and heat conductivity into account and studied effects of introducing latent heat and heat conductivity to the model. From those we clarified dependence of distribution of resist molecules on a substrate on latent heat of solvent. Moreover, we clarified why above characteristic dependence of distribution of resist molecules on latent heat appeared.
In Photomask Japan 2002 we reported a model of coating and drying process for the flat polymer (resist) film fabrication. The model models the process that polymer solution coated on a flat substrate by scanning technique, which is developed for LSI technologies at the next generation, is dried under reduced pressure. After above drying process, a resist film having a typical thickness profile that the edge of the film is thicker and just the region next to the edge's bump is thinner. Using the model, we numerically clarified the bumpy structure of the edge of a resist film depended on some essential parameters -- vaporization rate, diffusion coefficients coated solution thickness, intrinsic viscosity and so on. But the former model doesn't include the effects of a drop of temperature on the surface of a liquid film and the inside of it by latent heat and diffusion of heat by conductivity. But we cannot ignore these effects since a drop of temperature is essential in phase transition. And change of diffusion coefficients and viscosity by the drop of temperature has an important influence on the model.
In this paper, we will report the modified model taking a drop of temperature on the surface of a liquid film and the inside of it by latent heat and diffusion of heat by conductivity and change of diffusion coefficients and viscosity by it into consideration and numerical results using the model.
The coating procedure of polymer solutions by the scanning technique is developed for LSI technologies at the next generation, where a polymer solution as resists and inter-layer dielectric films is coated on a flat substrate, and then only the solvent is vaporized and removed, and finally the thin film is remaining there. In case of applying to the photo-lithography process, scan coating and its drying processes work together for astonishing flatness in 1% fluctuation range. When the coated polymer solution is dried under reduced pressure or vacuum, the thickness distribution of the resultant film should be accurately prospected and controlled by parameters. The film thickness is generally thicker at the edge and thinner inside from there than the average thickness. A typical thickness profile of a resist film is shown in Figure 1 . The phenomena are always observed, but have not been analyzed numerically. In this paper, we report a numerical model of the drying process of liquid film including polymers and give the essential parameters to the coating and drying processes. The parameters are focused on a vaporization rate, diffusion coefficients, coated solution thickness and intrinsic viscosity, which were calculated by simplified dynamical models of Langmuir's vaporization rate equation and Einstein relation at complex polymer solutions.
Access to the requested content is limited to institutions that have purchased or subscribe to SPIE eBooks.
You are receiving this notice because your organization may not have SPIE eBooks access.*
*Shibboleth/Open Athens users─please
sign in
to access your institution's subscriptions.
To obtain this item, you may purchase the complete book in print or electronic format on
SPIE.org.
INSTITUTIONAL Select your institution to access the SPIE Digital Library.
PERSONAL Sign in with your SPIE account to access your personal subscriptions or to use specific features such as save to my library, sign up for alerts, save searches, etc.