Analysis of Existing Thermodynamic Models of the Liquid Drop Deposited on the Substrate—A Sufficient Condition of the Minimum Free Energy of the System
Kenji Katoh
Round 1
Reviewer 1 Report
The paper describes a non-equilibrium thermodynamic model for the droplet deposition on a solid substrate. The paper is mathematically sound and gives a good insight on the problem of droplet spreading, by calculating the differences of the free energy between an initial and a final state. On the other hand, the text is poorly written and substantial improvements can be made on the clarity and use of English.
1. Starting for the use of English, I will only give some examples from the paper, although the list is far from exhaustive: Line number 32. "... used in the technique for ..." The word technique means "a way of carrying out a particular task", which makes no sense in this context. Line number 40. " ... incremental manufacturing methods (3D) ..." should be "... additive manufacturing methods (3D) ...". The word "verbally" = "by means of words" cannot be used in a sentence with citation. The authors obviously mean something else. Line number 137. " ... the first works concerned the issue ..." should be " ... the first works that addressed the issue ...". The word "zeroed" (as a verb) does not mean "set a term in an equation to zero", but "to adjust (an instrument or a device) to zero value".
All these are examples just from the 1st page; I strongly suggest that the paper should be edited to meet the standards, not just correct the aforementioned comments.
Going to the other recommendations:
2.Although the introduction gives a good historical background of the spreading problem, it is poorly written. The 1st paragraph of the introduction totally lacks citations, therefore the claims are not substantiated.
3.The paragraph starting at line number 90 "On the basis of thermodynamic consideration, ..." is impossible to follow and it is unclear how it is connected with the rest of the introduction.
4.The sentence at line number 140 " Compatibility of experimental observations (Youngs equation) significantly ..." does not make sense.
5.In equation (2), the term "Π" is neither justified nor helps the reader in any way. The quote ""other" energy" gives zero information and in my opinion should be either removed or explained.
6.As a suggestion, I would change the symbol θ (which in a lot of publications used for contact angles) to something more appropriate for length.
7.Line number 199, "In papers cited in the introduction ..." is a general statement and makes it impossible to understand which are these papers. The authors should report the exact citations.
8.Line number 219, " ... we do not discuss here the validity of this assumption ...". The sentence is sincere, but might confuse the reader. The author should add a citation where the reader can find a discussion on this assumption, or rephrase.
9. Line number 226, the sentence "In each case analyzed in literature, ..." needs citation.
10. Line number 255, sentence "A detailed description can be ...". A description of what? The authors should give a small idea to the reader what can be found in the Appendix.
11. Line number 256, sentence "After substituting equations ...". The substitution is performed on equation (7), not (13).
12. In equation (14), the initial free energy is given as F0, whereas in equation (1) as FI. The authors should use one of the two symbols throughout the manuscript.
13. Line number 267, sentence "... solved years ago [13.15].". The separator in citation should be a comma.
14. After equation (18) the numbering of the equations is repeated. For example the next equation is labeled as (11) instead of (19).
15. Line number 320, sentence "The equation that limits the area in which ..." is stated wrongly. The system of equations is not solved, but equation (20) is substituted to (16).
16. It would be helpful to the reader if the authors discuss what the dimensionless parameter G represents/scales.
17. Line number 352, "... increasing the contact angle." It is unclear which contact angle the authors refer to. Is the contact angle φ from the previous calculations or the equilibrium contact angle the droplet would have if γSLG = 0.
18. Line number 402-405. "In the case under consideration, ... in both systems". I do not understand the argument of the authors; why the energy has not changed? Also, is (25) pointing to the equation in the Appendix? How is this connected to the statement?
19. Line number 431-438. This paragraph is poorly written, it is unclear how gravity will affect the outcome of different experiments. It is unclear what the authors mean by "donation".
20. Line number 499-502. The sentence "It is also noteworthy ... three-phase tension line." is unclear.
Author Response
We would like to thank the reviewer for the great effort put into reviewing our manuscript and formulating many insightful comments.
1. Well, English is not our native language, and even using Thesaurus we can't guess the exact meaning of many words. The more we appreciate the importance of Your comments. We have included all of them by modifying the text of the paper. We just hope we did it correctly. We have also made
a number of corrections on the following pages.
2. This paragraph provides information indicating a link between the issue further discussed and what is already widely known and used. Indicates to a less advanced reader what the results of this work can be used for. This is a customary principle also applied in almost all the papers cited below. Only language corrections were introduced.
Lines: 32-42. “The phenomenon of wetting a solid substrate with liquids has been used in technical devices for centuries. Initially, the highest interest among engineers was the rheology of liquids and adhesion phenomena regarding the ability to create their uniform thin layers on solid surfaces – painting and lubrication then spray painting and coating. At the end of the twentieth century, due to the techniques of creating computer printouts, intensive work began on depositing on solid substrates and drops evaporation - inkjet printers. It seems that at the moment special attention should be paid to the shape and stability of drops created on solid surfaces, i.e. issues related to 3D printing, in a way preparing some background for the description of the behaviour of liquid drops created and flowing in microfluidic systems. It should be noted that additive manufacturing methods (3D) are increasingly applied by the industry, in turn, microfluidic devices are increasingly used in bio-, chemical and medical analyses.”
3. The paragraph has been partially reworded.
Lines: 90-104. “Based on thermodynamic considerations, the equation balancing the internal energy of a system composed of the following phases: gas, liquid and solid surface was formulated [12]. The energy of particular components of the system was taken into account, i.e. phases, interface surfaces, lines of contact of three phases, and even dividing points constituting the ending of these lines. The model thus formulated would be an excellent tool for determining the internal energy of a system consisting of a droplet deposited on a solid substrate both being in contact with the gas phase. Unfortunately, determining the necessary condition for thermodynamic equilibrium would require determining the exact differential of the internal energy of the system and equating it to zero. This in turn would require that the entropy values of all elements making up the system (phases, interfaces, three phase contact lines and dividing points) be identical. This condition significantly complicates calculations. Moreover, probably starting from the assumption that the real thermodynamic system always reaches the minimum of internal energy, the equations describing its sufficient condition were neither discussed nor formulated. However, this is the only way to verify the correctness of the assumptions introduced concerning the needs of thermodynamic equilibrium achievement of the system.”
4. The sentence has been redrafted.
Lines: 144-146.： “After introducing the dependence of tension in the three-phase tension line on the curvature of the wetted surface (Tolman equation) [27,28], the compatibility of the results of molecular computer simulations with the Young's equation was significantly improved.”
5. A brief explanation about "other" energies that may appear in the system has been added here. The presence of this term indicates a direct relationship between the issue under consideration and those in which these "other" energies are present.
Lines: 197-202.： “The twelfth component refers to "other" energy accumulated in the system or at the interfaces. This other energy will appear in case of acting on the system of forces of external fields, e.g. gravitational, electrostatic, magnetic, etc. Then in place of parameter one should write the appropriate mathematical expression describing a given type of interactions. However, there are no external force fields in the system under consideration, and therefore the term is equal to zero.”
6. θ was replaced by Λ.
7. It has been done.
Lines: 206-208.： “In the papers cited in the introduction describing thermodynamic models of settlement and spreading of liquid drops on the surface of a solid substrate [12-16,18,19], the volume of the liquid phase during the thermodynamic process is unchanged.”
8. It has been done.
Lines: 229-230.： “However it should be noted that we do not discuss here the validity of this assumption, but only introduce it because of its wide application e.g. [31].”
9. It has been done.
Lines: 236-237.： “In each case analyzed in literature, the axial symmetry of the drop is assumed [12-16,18,19,30-33].”
10. Two explanatory sentences have been added.
Lines: 265-268.： “In this case, the search for the free energy extremum of the system is changed into the search for extremum of the sum of surface energies and tension line occurring in the system. However, such a system has to be isothermal. In addition, the equations describing equilibrium and stationary systems remain unchanged. A detailed description can be found in the Appendix.”
11. Very accurate attention, because the order of described substitutions was incorrect. This has been corrected.
Lines: 270-272.：“After substituting equation (13) into equations (10-12) and entering the obtained dependencies into equation (7), and then into equation (1), which takes into account the simplifications used to derive equation (7), we get:...”
12. The postulated correction was introduced in the equations and text.
13. The separator has been changed.
Line: 282.： “One can only minimize equation (14) by looking for its minimum, but this problem was sketchy solved years ago [13,15].”
14. Despite the correct numbering in the manuscript, this problem appeared after submitting the paper. In the current version of the paper (manuscript revised) the numbering of equations is also correct, but we do not know if it will be kept during submission.
15. Equation (21) was obtained by adding equations (16) and (20) - Gauss method. Both the suggested method "by substitution" and the Gauss method are methods of solving systems of algebraic equations. For this reason, we did not change the text of the paper.
Lines: 335-337.：“The equation that limits the area in which the free energy of the system reaches the minimum from the rest can be obtained by solving the system of equations (16) and the following:
16. It has been done.
Lines: 409-411. “Parameter G is the ratio of the force acting along the wetted circumference being the three-phase boundary to the surface tension force acting on the surface of the spherical drop just before its deposition.”
17. The whole paragraph has been redrafted.
Lines: 372-381. “From the chart in Figure 2, it follows that for the zero value of the parameter , the course of the function given by equation (16) closely corresponds to the solutions of the Young equation, and in addition both conditions of the minimum free energy are met in the whole range of contact angle variability. This means that each of the solutions of this equation meets the conditions of thermodynamic equilibrium. As the value of this parameter decreases (for ), i.e. when the tension in three-phase tension line is stretching the circumference of the wetted area, the contact angle decreases. The strenght of the effect of changes in the parameter on changes in the contact angle is easiest to analyze using the assumption of While in the range of positive values of the parameter (), i.e. when the tension in the three phase tension line shrinks circumference of the wetted area, the contact angle increases with increase.”
18. The information in the next two sentences has been completed.
Lines: 436-440.： “In the discussed case the equilibrium system (classic thermodynamics) was replaced by a stationary system (non-equilibrium thermodynamics), however the equations defining free energy in both systems did not change in their integral (6) and (7) or differential (25) forms. Therefore, the properties regarding the changes of the analyzed free energy in processes will be the same in both systems.”
19. The information in the next two sentences has been completed.
Lines: 436-440. “In the discussed case the equilibrium system (classic thermodynamics) was replaced by a stationary system (non-equilibrium thermodynamics), however the equations defining free energy in both systems did not change in their integral (6) and (7) or differential (25) forms. Therefore, the properties regarding the changes of the analyzed free energy in processes will be the same in both systems.”
20. One sentence has been completed and redrafted and one sentence has been added.
Lines: 546-552. “It is also noteworthy that the conditions of liquid-vapour equilibrium can be met, and the lack of thermodynamic equilibrium of the system results from the inability to achieve a minimum by the sum of interfacial surfaces energy and energy accumulated in the three phase tension line. Exactly this reason is indicated by the analysis of equation (16). It has to be stressed that in such a case, failure to meet the thermodynamic equilibrium conditions prevents further analysis of the reasons and impacts of the system behaviour.”
Answers to all issues raised in the review are included in the attached file.
Author Response File: 
Author Response.docx
Reviewer 2 Report
1. This paper is a theoretical study on the equilibrium condition of the triple-phase contact line. In
addition to the classical equilibrium condition, the effects of line tension were mainly discussed. It
seems that the concluions include several meaningful results. Howere, the generality of the
parameter G related to the line tension is not clear. The authors do not provide sufficient
evaluation about the actural order of G and do not show any experimental results. In order to show
the generality of the theoretical results, the authors should discrss about the value of G with more
experimental results in literatures.
2. Others: Some equation numbers are wrong. Please correct.
Author Response
1. Based on the experimental results taken from the literature, the range of variability of parameter
G values was estimated. It was also signaled, citing Erbil's work [34], the discrepancies of the
measured tension values in the three phase tension line in macroscopic systems and those determined
on the basis of molecular simulations for nanometer droplets. Lines: 359-365.
“In the literature [34], there is a very large discrepancy between the values of the three phase tension line
calculated theoretically based on intermolecular interactions and those measured in experiments. However, the
widespread use of equation (17) to approximate the results of experiments prompts the use of experimental rather
than theoretical values. In our considerations, we take into account the droplet volume greater than 0.01 [μL]
and the tension values at the three-phase tension line given in the papers, e.g. in [13, 14], and then the factor can reach values up to approximately ± 5.”
The corrected version of the paper gives the physical meaning of the G parameter. Lines: 409-411.
“Parameter is the ratio of the force acting along the wetted circumference being the three-phase boundary to
the surface tension force acting on the surface of the spherical drop just before its deposition.”
2. The numbering of the equations was distorted during the submission of the paper. The current
version of the manuscript is correct. However, we are not sure if submitting the paper again will not
cause undesirable changes.
Author Response File: Author Response.docx
Reviewer 3 Report
1. The authors use the principles of non-equilibrium thermodynamics in order to examine the conditions under which a deposited droplet reaches its minimum free energy.
The manuscript is well-written and I believe will be interesting for the readers of the journal.
The authors should also include a comment regarding their future work (e.g. any possibilities of extending their model on geometrically structured solid substrates).
Author Response
1.We would like to thank the reviewer for his/her positive and insightful comments on the manuscript.In line with ethics, I always keep my promises. However, given the complexity of academic life,
I cannot define very strictly the issues I will deal with. Therefore, instead of presenting my research plans, I put a short paragraph in the paper indicating the issues in which the results of this work may be useful. Lines: 502-506. “The solutions shown in the paper may be a limiting solution for systems in which the surface of
a solid substrate is modified (roughness, texture) to improve or to deteriorate its wettability [36,37]. However, such changes in wettability have an impact on the efficiency of microfluidic devices
[38, 39], which means that a change in the technology for producing microfluidic chips will have an impact on the quality of their operation.“
Author Response File: Author Response.docx
Round 2
Reviewer 1 Report
The authors implemented most of my comments, except two:
The first omission I find mildly important: the authors did not include any citations in the 1st paragraph of the introduction to back up their claims. Although they responded that the information is "widely known and used", I doubt that the "less advanced reader" will know about it. Moreover, the fact that the call this practice "a customary principle" is just absurd.
The second omission is the use of English and I find it important. Although the authors did try to implement my comments (by the way, they substituted the word "verbally" with "in words", but still cite a written document), the text has numerous mistakes. I understand that they are not native speakers, but this not an excuse; the reader should be able to understand the text.
Reviewer 2 Report
The authors respond appropriately to the reviewer's comments. I recommend that the paper be published in the journal.
