Pressure dependence of surface tension of polymer melts under high vacuum

Domain: chemistry
Published: 2026-03-03
Editorial outcome: major_revision

Abstract
The surface tension of polymers under ambient or high-pressure conditions has been
extensively studied but is less explored under high vacuum. Here, the effect of air pressure on the
surface tension of polymer melts is studied by using a home-built apparatus. While previous
studies showed that the surface tension of most polymers decreases linearly when the temperature
or pressure is increased, we show that the surface tension of polymer is found to exhibit an
anomalous behavior: The surface tension remains a decreasing function of the temperature, but
increases with increasing air pressure. The measured surface tension of several polymer samples
is well described by the Hill equation in the pressure range of 10−4 to 105 N/m2. The implication
of this behavior on the air-polymer interaction is discussed within the context of air adsorption

Introduction
The surface tension (𝛾) of polymer melts is an important property of polymeric materials and
plays a key role in numerous engineering processes, such as polymer blending1, polymer foaming2,
wetting3, and particles or fiber dispersion in polymers4. Naturally, the surface tension of polymers
depends on the thermodynamic conditions, commonly specified by the temperature (𝑇) and
pressure (𝑃). It is well-established that the surface tension of most polymer melts decreases slightly
with the temperature, following approximately a linear relationship, at ambient pressure of the
order of 105 Pa5-10. Similarly, the pressure dependence of 𝛾, determined under high-pressure
conditions, was observed to decrease linearly when the pressure is increased at a constant
temperature11-18. The decrease of 𝛾 with increasing 𝑃 is attributed to either increased gas
dissolution or the reduction of density difference between the two phases across the interface11-18.
However, the pressure dependence of 𝛾 under high vacuum conditions has not been studied prior
to the current work.

In contrast to the abundant studies under ambient or high-pressure, the phase behaviors of
polymers and their surface tension under high vacuum conditions have attracted much less
attention, largely due to the difficulty of precisely controlling pressure to reach high vacuum
conditions. Recently, it has been discovered that placing diblock copolymers under high vacuum
conditions could lead to a drastic orientation change from parallel to perpendicular for the self-
assembled cylindrical nanostructure19. The origin of this interesting discovery could be attributed
to a much-reduced surface tension difference between the different blocks, i.e. polystyrene (PS)
and polydimethylsiloxane (PDMS) or poly(L-lactide) (PLLA), under high vacuum20,21. While it is
natural to suggest that the surface tensions of the polymers depend on the pressure, a systematic
study, quantitatively or even qualitatively, of the effect of air pressure on the surface tension of
polymers has been lacking. Here, we fill this gap by carrying out a systematic study of the effect
of air pressure on the surface tension of polymer melts under high vacuum. Specifically, we
designed and constructed a homemade vacuum oven (Supplementary Figure. 1) that enabled us to
achieve a high vacuum and measured the surface tension of polymer melts by the pre-coated
capillary height method. It is noted that several techniques, such as pendant, sessile, spinning drop,
etc. could be used to obtain the surface tension. These methods have the advantage of requiring a
minimum amount of sample and are applicable to various cases22-24. However, their performance
and sensitivity significantly decrease when the drop shape becomes nearly spherical25,26. In the
current study, the pre-coated capillary technique was chosen due to its simplicity and reliability20.
We carried out surface tension measurements on several representative polymer samples,
including polyethylene glycol (PEG), polyisoprene (PI), polypropylene (PP), PS, and PDMS, with
a wide range of surface tension values under ambient conditions. A very surprising observation
from our study is that, in contrast to the pressure effect on the surface tension under ambient or
high-pressure, the surface tension of all the polymers exhibits a significant decrease when the
pressure drops below 103 N/m2. This anomalous behavior of the polymer surface tension must be
related to the polymer-air interactions. This new finding reveals interesting physics at the air-
polymer interfaces and, furthermore, sheds light on potential applications of designing polymeric
materials for nanopatterning technologies within nano-microelectronic manufacturing systems
(MEMS).

Results
Temperature Dependence of Surface Tension
To explore the temperature dependence of surface tension for polymeric melts, systematic
measurements were carried out using a homemade vacuum oven apparatus. As a validation of our
experimental method, the surface tension of the polymer samples (shown in Fig. 1) was obtained
as a function of temperature in the range from 170 to 220°C, under ambient pressure (~105 N/m2).
It is noted that the surface tension data for PS and PDMS were reported in our previous work20.
The data for PI was not available for temperatures over 200°C due to thermal degradation of the
sample. In agreement with the literature5-10, the measured surface tension shown in Fig. 1 for the
five polymers (PEG, PS, PI, PP, and PDMS) exhibits a slight linear decrease when the temperature
is increased, with a slope in the range of -0.08 to -0.12. The agreement between the results from
the current study and the literature validates our experimental methods.

Pressure Dependence of Surface Tension
Several previous studies have demonstrated a small reduction in surface tension at high
pressure and temperature. However, there has been less focus on the low-pressure region. To fill
this gap, we carried out measurements of polymer surface tension under low-pressure conditions.
The homemade oven allowed us to access low-pressure conditions, thus enabling the measurement
of the surface tension under high vacuum conditions. Specifically, surface tension of several
sample polymers was obtained in the pressure range from atmospheric pressure (105 N/m2) to 10-
4 N/m2. The measured surface tension of the samples at a temperature of 200°C are presented in
Table 1, revealing a clear decrease in surface tension with decreasing air pressure. Interestingly, a
drastic reduction in surface tension is observed as the pressure drops below atmospheric levels
(105 N/m2). Since the pressure change was over 8 orders of magnitude, a linear plot of our data
compresses most of the points to a vertical line. Instead, it is the best to use log-linear plot to
demonstrate the change of the measured surface tension. The data points and their best fitting by
Hill equation as shown in Fig. 2. It is important to mention that the trend of the surface tension
change is qualitatively different from the pressure effect under high-pressure conditions, where a
slight decrease of 𝛾 is observed when the pressure is increased11-18. Note that the low- and high-
pressure regimes involve distinct mechanisms. At low pressure, surface tension changes are
primarily governed by modifications in polymer–air interactions, as captured by the lattice model.
At high-pressure, the decrease in surface tension arises from gas dissolution or the reduced density
difference across the interface11,13.

To examine the generality of the observed behaviour, we measured the surface tension of two
polymers, PEG and PDMS, with different molecular weights under high vacuum conditions. These
two polymers were chosen because they possess the highest (PEG) and the lowest (PDMS) surface
tension under ambient conditions. Note that the PEG with a molecular weight (MW) of 20000
g/mol is referred to as PEG HMW and PEG with MW of 10000 g/mol molecular weight is referred
to as PEG LMW. Also, low MW (21,000 g/mol) PDMS is referred to as PDMS LMW and high
MW (38,000 g/mol) PDMS is referred to as PDMS HMW. As shown in Fig. 3, the molecular
weight of the polymers has a negligible effect on the surface tension. The observed pressure
dependence of surface tension is consistent across polymers of different molecular weights,
indicating that polymer chain length or PDI does not play a dominant role. Instead, the trend
primarily reflects intrinsic polymer-air interfacial interactions, suggesting that the pressure
dependence should be universal to polymers. It is plausible that the observed pressure dependence
of surface tension is not polymer-specific. Rather, it is the result of surface-air adsorption. In what
follows, we construct a theoretical model to understand and explain the observed temperature and
pressure dependence of the surface tension.

Theoretical Model
To understand and explain the experimental observations, a theoretical model of the surface
tension of a condensed sample, such as a polymer melt, surrounded by air molecules is desirable.
For a system composed of a substrate and a gas, the surface tension could be divided into two
components, the surface tension of the neat material 𝛾 and the contribution from interaction with
0
the air molecules, ∆𝛾. The reduction of the surface tension in high vacuum (low pressure) region
is due to air-polymer interaction and that could be estimated by using a lattice gas model. We
assume that half of the lattice is occupied by polymeric species (𝑃), while the other half of the
lattice is occupied by gas (air) molecules (𝐴) or empty (vacuum) (𝐵). We use a simple cubic lattice
in the following discussion, although any type of lattice would be equally valid. We assume
nearest-neighbour interactions between the different species. For a surface in the xy plane, the
surface area is 𝑆 = 𝑁𝑎2 where 𝑁 is the number of lattice sites on the xy plane and 𝑎 is the lattice
constant. The total energy between the polymeric species and the lattice gas is given by,
𝐸 = 𝑁 𝜀 +𝑁 𝜀 , (1)
𝑃𝐴 𝑃𝐴 𝑃𝐵 𝑃𝐵
where 𝑁 and 𝑁 are the number of nearest-neighbor polymer-air and polymer-vacuum bonds,
𝑃𝐴 𝑃𝐵
respectively, and 𝜀 and 𝜀 are the corresponding interaction energies. We use a mean-field
𝑃𝐴 𝑃𝐵
approximation to estimate the number of nearest-neighbor bonds. Specifically, we assume that the
air concentration is given by 𝜙, then the average number of nearest-neighbor bonds are given by,
𝑁 = 𝜙𝑁 and 𝑁 = (1−𝜙)𝑁. Therefore, the total surface energy becomes,
𝑃𝐴 𝑃𝐵
𝐸 = 𝑁𝜙𝜀 +𝑁(1−𝜙)𝜀 = 𝑁[𝜀 +𝜙(𝜀 −𝜀 )] (2)
𝑃𝐴 𝑃𝐵 𝑃𝐵 𝑃𝐴 𝑃𝐵
Using the expressions of the total surface energy and surface area, the surface tension that is the
surface energy per unit area is obtained as,
𝛾 = 𝐸 = 𝜀𝑃𝐵+𝜙(𝜀𝑃𝐴−𝜀𝑃𝐵 ) (3)
𝑆 𝑎2
We now introduce the polymer-vacuum surface tension, 𝛾 = 𝜀𝑃𝐵,
and the differential interaction
0 𝑎2
energy ∆𝜀 = 𝜀 −𝜀 , the surface tension is now given by,
𝑃𝐴 𝑃𝐵
𝛾 = 𝛾 + 𝜙 ∆𝜀 (4)
0 𝑙2
where 𝑙 is a length scale characterizing monomer-monomer distance, ∆𝜀 quantifies the difference
of air-polymer and vacuum-polymer interaction, and 𝜙 is the concentration of air molecules at the
polymer surface that is the key property determining the pressure-dependence of the surface
tension.
The simplest assumption is that there is no adsorption of air molecules. In this case, the air
concentration is given by the ideal gas law, 𝜙 = 𝑁 𝑣0 = 𝑣0𝑃 , where 𝑣 is the volume of one gas
𝑉 𝑘𝐵𝑇 0
molecule. Using this expression obtained,
𝛾 = 𝛾 + 𝑣0∆𝜀 𝑃 (5)
0 𝑙2 𝑘𝐵𝑇
This result predicts that the surface tension is an increasing function of the pressure, 𝑃, and a
decreasing function of the temperature, 𝑇. This predicted behavior is in qualitative agreement with
our experimental observations. However, this simple model does not provide an explanation of the
observed relationship reported above. Therefore, a more sophisticated model for air adsorption is
needed. When air adsorption to the polymer surface occurs, the surface concentration of air
molecules becomes larger than that predicted by the ideal gas model. The reduction of surface
tension in the high-pressure region is primarily due to the increase in the dissolution of the gas
molecule (solubility) with the increase in pressure12,14-18. The drop in surface tension in high-
pressure can also be explained in terms of density which is mostly attributed to the reduction in
density difference between the gas molecule and polymer at the interface11,13. On the other hand,
the behaviour of the surface tension in the high vacuum region must be due to different
mechanisms. Here, we propose that the reduction of surface tension with the decrease of pressure
could be attributed to the adsorption of air molecules to the surface. We assume that the air
adsorption is described by the Hill’s function32,33,
𝜙 = 𝑃𝐻 (6)
𝑃𝐻+𝑃𝐻
0
Here 𝐻 is the Hill coefficient and 𝑃 represents a threshold pressure at which 𝜙 = 1/2. The
familiar Langmuir adsorption isothermal (𝜙 = 𝑃/(𝑃 +𝑃)) is recovered when 𝐻 = 1. The Hill
0
equation is used widely in biochemistry to describe the ligand binding, which is similar to the
adsorption process. The Hill coefficient is commonly found to be larger than 1 in biologically
systems, indicating positive corporative binding. The case with 𝐻 < 1 corresponds to negative
cooperative binding. Since the polymer and air molecules are not strongly interacting, a small Hill
coefficient is not an unreasonable assumption. Using the Hill equation, we can write the surface
tension as,
𝛾 = 𝛾 +(𝛾 −𝛾 ) (𝑃/𝑃0 )𝐻 (7)
0 ∞ 0 1+(𝑃/𝑃0 )𝐻
Here 𝛾 , 𝛾 and 𝑃 are fitting parameters for a fixed value of the Hill coefficient 𝐻. 𝛾 is surface
0 ∞ 0 0
tension in vacuum (P=0) and 𝛾 represents the surface tension where air adsorption is effectively
∞
saturated. We have tested different values of 𝐻 and found that a small 𝐻 < 1 gives good fitting to
all the data. After some exploration, we decided to use 𝐻 = 0.2 that gives good fitting to all the
data. The fitting was carried out by using the NonlinearModelFit function of Mathematica. It is
found that all the data could be well described by this equation with 𝐻 = 0.2 and the fitting
parameters are given in Table 2. Such negative cooperative binding (H = 0.2) is generally observed
in ligand-protein binding.35 It is interesting to note that for finite value of 𝑃/𝑃 and small 𝐻, the
0
Hill function can be approximated as 𝜙 ≈ 1 (1+ 𝐻 ln𝑃/𝑃 ) , therefore the observed surface
0
2 2
tension data could be described by a linear function of ln𝑃.Note that the parameter 𝑃 in our model
0
as shown in Equation 7 can be temperature-dependent. When the Hill function is used, the
temperature-dependence of the surface tension is embedded in the parameter 𝑃 in our model.
0
The temperature-dependence shown in Figure 1 could be explained by using the approximate
expression 𝜙 ≈ 1 (1+ 𝐻 ln𝑃/𝑃 ) .
0
2 2
As compared to other polymer systems, the PI sample shows a different behaviour in that the
fitting parameter 𝑃 is much larger than that for all other samples. We speculate that this might be
0
attributed to thermal degradation of PI since its degradable temperature is very close to the
experimental temperature (~200 °C). The degradation could lead to subtle changes in surface
composition or reactivity, thus resulting in a rather different fitting parameter. Overall, the quality
of fitting is good for the data points over the range of the pressure.
The fitting formula allows us to display all the data on a universal plot, if we express the
surface tension using the following scaled form,
𝛾−𝛾0 = (𝑃/𝑃0 )𝐻 (8)
𝛾∞−𝛾0 1+(𝑃/𝑃0 )𝐻
The scaled data and the Hill function are shown in Fig. 3. It is very interesting to see that all the
data points fall on a master curve. The fact that all the data is well described by one master curve
indicates that the physics governing the behaviour of the surface tension as a function of air
pressure at low pressure region should be universal, independent of the polymer samples. It is
noted that air is a homogeneous mixture of N , O and Ar. In our simple model, the contribution
2 2
of different interactions of those gas molecules with polymer is represented by an averaged
interaction parameter. Certainly, the model could be improved by performing control experiments
to differentiate the interaction between individual molecule and polymer. However, the behaviors
with respect to the pressure dependence of surface tension for polymer melts under vacuum should
be the same. We would like to mention that Herminghaus et al. observed that the pressure effect
on polymer thin film rupture and shown that the effect is insensitive to the gas molecules34. When
the experiments were carried out under vacuum conditions below roughly 1 mbar, no rupture
occurred while the substrate was properly cleaned beforehand. In contrast, at pressures of 10 mbar
or above, hole formation was consistently observed, independent of the gas type. Pressures at or
below 1 mbar effectively inhibited rupture in all cases. Because thin film rupture is largely driven
by the surface tension of the polymer, their observations imply a sharp decrease of polymer surface
tension when the gas pressure is decreased to below 1 mbar, independent on the gas molecules.
These observations are consistent with our results on the pressure-dependence of polymer surface
tension. Although many details of the air-polymer interactions are ignored, the adsorption model
with Hill function does provide an adequate description of the experiments.

Methods
Materials.
The polyethylene glycol (Mw = 20,000 g/mol, catalogue no. A17925.30) from Thermo
Scientific and polyethylene glycol (Mw = 10,000 g/mol, product no. P6667) from Sigma-Aldrich
used in this study were commercially available products. Polystyrene (Mw = 20,000 g/mol, PDI =
1.06), polyisoprene (Mw = 19,000 g/mol, PDI = 1.03), and polydimethylsiloxane (Mw = 21,000
and 38,000 g/mol, PDI = 1.09) were synthesized in the laboratory. Polypropylene (Mw = 12,000
g/mol, product no. 428116) used was a commercially available product and obtained from Sigma-
Aldrich.
The polymer samples, whether commercially obtained or prepared in the laboratory, were
used without additional processing such as solution casting or heat pressing. The samples were
either in powder form (e.g. PEG, PS, PP) or in viscous liquid-like form (e.g. PI and PDMS), and
were placed directly into clean glass vials. Each sample was placed in a clean glass vial along with
the suspended capillary tube and kept in a homemade vacuum oven. The rotary and diffusion pump
were operated overnight to achieve a high degree of vacuum in the chamber. Following this, the
oven was heated to the desired temperature while maintaining vacuum conditions. This process
enabled thermal annealing under high vacuum prior to testing.

Measurement of surface tension for polymeric melt
The surface tension measurements of representative polymeric systems were carried out using
the capillary rise method under zero-contact-angle conditions (See Supplementary information,
Pre-coated Capillary Height Method section for details). The dependence of surface tension on
thermodynamic variables like temperature and pressure was studied through experiments
conducted using a homemade vacuum oven apparatus, which was designed to precisely control
temperature and pressure conditions (See Supplementary Figure 1. for details). Five commonly
used polymers, including PDMS, PP, PI, PS, and PEG, ranging from low cohesive density to high
cohesive energy density, were employed in this study, and the values of surface tension at ambient
conditions are shown in Supplementary Table 1.

Data availability. The data that support the findings of this study are provided with this paper
and in the supplementary information. All additional data are available from the corresponding
author upon request.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "In the manuscript, the authors measured the interfacial tension using the capillary height method. For a broad journal such as Nature Communications, the analysis of interfacial tension for the technique should be expanded on.\n1. The authors propose a model that they claim captures the physics of changing interfacial tension as a function of pressure, but from what I could tell, they only used linear fits with two data points per line (Fig 5). If I am in error here, and more data points are used, the plot should be changed for clarity, since each line appears only to have 2-3 points over a broad pressure range. Furthermore, the cohesive energy of the polymers used is known; how do the cohesive energies extracted from this fit compare with literature values?\n2. In Figure 2, the authors report an abrupt decrease when the air pressure drops below 103 Pa. However, only a single data point is provided above this threshold. Additional data points in that pressure range to support the observed trend.\n3. Could the authors clarify the specific ranges defined as low pressure and high pressure in this study?\n4. It is also suggested that the authors consider collecting surface tension data of the polymer melts under high pressure using the same method.\n5. On page 5, the authors state: “A very interesting observation from this plot is the descending order of 1.65, 1.51, 1.25, 1.21, and 0.41 for PEG, PS, PI, PP, and PDMS, respectively.” Could the authors provide an explanation for this observed trend among the different polymers?\n6. According to the fitting results (Table 1), the K value is 4 and appears to be invariant across different polymer types. Since the Langmuir constant (K) reflects the extent of interaction between the adsorbate and the surface, could the authors clarify whether the K value in Table 1 was derived from fitting or assumed? Additionally, could the authors explain why the K value remains constant across different polymers?\n7. The authors suggest that the observed pressure effect should not be specific to polymers and should be universally applicable to any physiochemical systems. However, they do not provide any measurement data for other material types, such as ionic liquids, surfactants, or rigid molecules. Moreover, the lattice model and Langmuir isotherm employed appear overly simplistic. Polymers, particularly block copolymers, exhibit surface inhomogeneities and non-one-dimensional interfaces due to fluctuating chain ends, making the Langmuir adsorption isotherm an inadequate approximation for such systems.\n8. The authors propose a model that they claim captures the physics of changing interfacial tension as a function of pressure, but from what I could tell, they only used linear fits with two data points per line (Fig 5). If I am in error here, and more data points are used, the plot should be changed for clarity, since each line appears only to have 2-3 points over a broad pressure range. Furthermore, the cohesive energy of the polymers used is known; how do the cohesive energies extracted from this fit compare with literature values?\n9. The research is of broad interest, but I think some work needs to be done to validate their model of lowering interfacial tension due to gas adsorption.\nSome further comments:\n1. Page 5; last two sentences of 1st paragraph – “This could be presented better. The way this is worded now sounds like they are saying more than just the temperature change is varying the surface tension even though the pressure is being kept constant. I think they mean to say that the surface tension could be changed more dramatically by changing other factors, such as pressure.”\n2. Page 5; middle of 2nd paragraph – “Should use a log scale for pressure in this graph to better illustrate this point because most of the important data points are overlapping significantly and the pressure at which the decrease happens isn’t obvious.”\n3. Page 5; second to last sentence of 2nd paragraph – “Okay, so they do this in Figure 3 but why even include figure 2 then since it’s so unclear?”\n4. Page 5; last sentence of 2nd paragraph – “What does this conclusion even mean? There are no units, and the plot doesn’t clearly reflect these values.” – I see now that they probably mean the slope values, but they don’t draw any conclusions about the significance of this decrease here or anywhere else in the manuscript.\n5. Page 6; middle of 1st paragraph – “Normally I’d ask what the pdi is for each polymer here, but, since we’re dealing with a melt sample and the trend remains unchanged, perhaps it isn’t the most relevant. Depending on what’s said about the surface-air adsorption part of their theoretical model, the pdi may or may not be important to consider.”\n6. Page 9; middle of 1st paragraph – “Is this for the same scenario (high-pressure) as the previous sentence?”\n7. Page 10; end of 1st paragraph – “Back to my earlier point about pdi, perhaps pdi is contributing to this difference here since lower MWs preferentially locate at the surface and you cite air-polymer interactions at the interface to be the main contribution to the surface tension reduction. Also, what specifically is contributing to the differences in the slope values for each polymer?” – They should also mention why they think PI is different from the rest of the polymers.", "reviewer_2": "The manuscript presents an interesting experimental study of the surface tension of polymer melts at low pressures. A threshold pressure is observed below which surface tension and pressure feature a strong positive correlation. To rationalize the existence of the pressure threshold the authors introduce a Langmuir isotherm model that justifies the existence of a critical pressure threshold with the hyperbolic shape of the isotherm.\nOverall this is an interesting, innovative and well-executed study. Only few experimental studies of the surface tension of liquids (and polymers) have been reported in the low-pressure regime, thus there is potential that this study to stimulate follow-up work. It is noted that the performance of materials in low-pressure environments is increasingly receiving attention due to novel application demands, for example, in space exploration.\nHowever, while I view this study very favorably overall, I am not compelled to recommend publication in Nature Communications. Although the study is pioneering in character, the scope is rather niche. Although the results could stimulate follow-up research in both fundamental and application-related areas for polymer materials, the focus of such activities is not elaborated in the proposal. There are also some 'loose ends' which leave the reader puzzled. For example, what is the role of chemistry on the observed differences between the different polymeric systems, molecular characteristics (for example, differences in dispersity indices) between the polymers are not mentioned. I did not find a description of the actual sample preparation process. For example, were materials solution processed or heat-pressed, where the materials, were materials conditioned by vacuum annealing prior to testing? It would have been interesting to evaluate for one chemistry the role of chain end density, for example, by variation of the molecular weight.\nIn summary, this is a high quality innovative work that deserves to be published in a specialized journal on polymer science."}, "author_response": ""}, {"round": 2, "reviewer_comments": {"reviewer_1": "The authors have made an interesting observation that the apparent interfacial tension decreases with decreasing pressure and that the reduction can be mapped on to a function put for by Hill to describe the behavior of receptors in biological systems. Here, the authors speculate that the surface of the molten polymers adsorb air molecules, and it is those adsorbed molecules that give rise to the observed variation in the apparent interfacial tension. Unlike biological systems where receptors have specific interactions, any interactions with the air molecules are left to the reader to decide. Air is a mixture of gases (78% N2, 21% O2 and 0.93% Ar). The authors make no attempt to discern between these and treat air as a single component gas, which it is not, nor do the authors perform any control experiments to differentiate between them. Certainly, there must be a difference between the adsorption of these different gases.\nThe master curve that the authors describe simply says that the functional form of the reduction is the same, suggesting, though not proving, that origin and mechanism of the adsorption is similar. Normalization of the data at the 50% change point, P0, and the range in the change of   ensures that the data will overlap. So, I am less impressed with the master curve and am still not informed as to the origin of the observation. Is it adsorption? Perhaps, but I would have expected some dependence on the chemical composition or the density of the polymer (free volume or concentration of chain ends at the surface or chain orientation at the surface or the interfacial tension of the specific polymer). None of these appear to be the case. So, what is it? The adsorption of interest is only at the surface, at least that is what will change the interfacial tension, but the surfaces of the different polymers are chemically different, have densities that are different and dynamics that will be different, but the measurements done here are insensitive to these differences. As I said in my earlier review, “I think some work needs to be done to validate their model of lowering interfacial tension due to gas adsorption.”, and I still feel that is the case. This work originated in the research that Ho had done on the orientation of block copolymer domains and there I could understand a preferential adsorption of any of a variety of vapors (gases, water vapor, or other gas) that could mediate the surface energies of the components. Yet, given the arguments the authors present in this manuscript, I am not certain that holds.\nConsequently, even though I find the observations interesting, I do not feel that this manuscript has the scientific rigor for publication in Nature Comm. I strongly encourage the authors to pursue this further, since there will be some interesting physics underpinning the observed behavior, but I just do not see that in this manuscript.\nAs an aside, I do want to point out that S. Herminghaus and coworkers performed dome detailed studies on polymer thin films stability, dewetting, etc. where they also observed difference when the homopolymers and mixtures showed different behavior under vacuum or different gas atmospheres which they attributed to variation in the interfacial tensions and it would be worthwhile for the authors to look at his work.", "reviewer_2": "In my original review, I noted several points that needed revision prior to publication of the manuscript. The authors have expanded significant efforts to improve the quality of the original submission and I find that the revised manuscript comprehensively addresses all the points I raised (with the exception of principle suitability for Nature Commun). I commend the authors for significant extension of the data range, in particular, the study of M-dependence. I also appreciate the extended discussion of sample preparation procedures. While I remain skeptical about the suitability of such a narrow theme for a broad audience journal, the principal quality is now (in my opinion) on par with work that would generally be considered suitable for high impact broad audience journals. I have no further changes to suggest.", "reviewer_3": "Please see review attachment"}, "author_response": "Responses to Reviewers’ Comments & Questions\nReviewer #1:\nComments: In the manuscript, the authors measured the interfacial tension using the capillary\nheight method. For a broad journal such as Nature Communications, the analysis of interfacial\ntension for the technique should be expanded on.\nReply: Thanks for the suggestion; as suggested, the analysis of interfacial tension for the\ntechnique has been expanded.\nQ1. The authors propose a model that they claim captures the physics of changing interfacial\ntension as a function of pressure, but from what I could tell, they only used linear fits with two\ndata points per line (Fig 5). If I am in error here, and more data points are used, the plot should\nbe changed for clarity, since each line appears only to have 2-3 points over a broad pressure\nrange. Furthermore, the cohesive energy of the polymers used is known; how do the cohesive\nenergies extracted from this fit compare with literature values?\nA1: We thank the Reviewer for this helpful comment. The fitting of Figure 5 was done for all\nthe data points we had. However, most of the data points were invisible due to our poor choice\nof using a linear plot. Since the pressure change was over 8 orders of magnitude, a linear plot\ncompresses most of the points to a vertical line. For such large range of pressure, it is the best\nto use log-linear plot to demonstrate the data points and fitting. Furthermore, we have redone\nthe data fitting using a new model, the Hill equation, on an expanded data set. Details of the\nfitting function are given below. All the data points and the corresponding fitting curves are\nshown in Figure R1 (Figure 2 in the revised manuscript) that is reproduced below. The plot is\nthe fitting of all data points. However, the data for PI and PDMS appear to deviate slightly\nFig. R1. Log-linear plot of all the data point and the fitting curves.\n1\nfrom the general trend observed. Overall, quality of fitting is good for the data points over the\nrange of the pressure.\nBased on the lattice model, the surface tension has the form 𝛾 = 𝛾 +𝛼𝜙, where 𝜙 is the\n0\nfraction of adsorbed air molecules on the surface. Instead of the Langmuir equation, we assume\nthat the adsorption is described by the Hill’s equation,\n𝑃𝐻\n𝜙 = .\n𝑃𝐻 +𝑃𝐻\nfamiliar Langmuir adsorption formula (𝜙 = 𝑃/(𝑃 +𝑃)) is recovered when 𝐻 = 1.\nThe Hill\nequation is used widely in biochemistry to describe ligand binding, which is similar to an\nadsorption process. The Hill coefficient is commonly found to be larger than 1 in biologically\nsystems, indicating positive corporative binding. The case with 𝐻 < 1 corresponds to negative\ncooperative binding. Since the polymer and air molecules are not strongly interacting, a small\nHill coefficient is not an unreasonable result. Using this equation, we can write the surface\ntension as,\n𝑃𝐻 (𝑃/𝑃 )𝐻\n0\n𝛾 = 𝛾 +𝛼 = 𝛾 +(𝛾 −𝛾 ) .\n0 𝑃𝐻 +𝑃𝐻 0 ∞ 0 1+(𝑃/𝑃 )𝐻\n0 0\nHere 𝛾 , 𝛾 and 𝑃 are fitting parameters for a fixed value of the Hill coefficient 𝐻. 𝛾 is\n0 ∞ 0 0\nsurface tension in vacuum (P=0) and 𝛾 is surface tension at infinite pressure (P=∞). We have\n∞\ntested different values of 𝐻 and found that a small 𝐻 value gives good fitting to all the data.\nAfter some exploration, we decided to use 𝐻 = 0.2 that gives good fitting to all the data. The\nfitting was carried out by using the NonlinearModelFit function of Mathematica. It is found\nthat all the data could be well described by this equation with 𝐻 = 0.2 with the fitting\nparameters shown in Table R1 (Table 2 in the revised manuscript). An example of the data\nfitting is shown in Fig. R2 for the case of polystyrene, while all the data points and fitting\ncurves are given in Fig. R1 (Figure 2 in the revised manuscript). It is interesting to note that\nthis fitting formula allows us to display all the data on a universal plot, if we express the surface\ntension using the following scaled form,\n𝛾 −𝛾 (𝑃/𝑃 )𝐻\n0 0\n=\n𝛾 −𝛾 1+(𝑃/𝑃 )𝐻\n∞ 0 0\nThe scaled data and the Hill function are plotted in Fig. R3. All the data points are on a master\ncurve.\nFig. R3. Fitting of surface tension versus pressure plot for PEG, PS, PI, PP, PDMS, and PEG\nLMW using the proposed model to obtain master curve.\nTable R1. Fitted parameters for all data points.\n𝛾 𝛾 𝑃\n0 ∞ 0\nPEG 16.2 35.2 0.10\nPEGLmw 13.5 31.9 0.03\nPS 11.6 27.5 1.15\nPI 13.5 28.1 226.2\nPDMS 9.2 13.1 9.6\nPDMSLmw 7.6 13.4 1.19E-3\nPP 9.8 23.0 0.311\nthe behavior of the surface tension as a function of air pressure at low pressure region should\nbe universal, independent of the polymer samples. The revised theoretical model was added on\npages 8 and 9 in the revised manuscript.\nRegarding the extracted cohesive energy values, comparison with literature data is\nchallenging because surface tension values at high temperatures and low pressures are\ngenerally not available. This limits the extent of direct comparison with reported literature\nvalues.\nQ2. In Figure 2, the authors report an abrupt decrease when the air pressure drops below 103\nPa. However, only a single data point is provided above this threshold. Additional data points\nin that pressure range to support the observed trend.\nA2: We thank the Reviewer for the comment. As suggested by the Reviewer, additional\nexperiments were carried out to have data points in that pressure range to support the observed\ntrend, as described in A1. Specifically, additional experiments were carried out at P=17500 Pa\nfor PS, PP and low-molecular weight PEG. These data points are consistent with the previous\ndata points and were added to the data fitting.\nQ3. Could the authors clarify the specific ranges defined as low pressure and high pressure in\nthis study?\nA3: Thank you for the comment. In this study, we primarily focus on the low-pressure region,\ndefined as 10³ Pa to 10⁻⁴ Pa. This is the range where we observed significant changes in surface\ntension. The high-pressure region (above atmospheric pressure) was not the focus of our\ninvestigation and have been studied extensively before (See Li, H. et al. Ind. Eng. Chem. Res.\n43, 509-514 (2004); Park, H. et al. Ind. Eng. Chem. Res. 45, 1650-1658 (2006). etc).\nQ4. It is also suggested that the authors consider collecting surface tension data of the polymer\nmelts under high pressure using the same method.\nA4: Thanks for the comment. In this study, the homemade vacuum oven is designed using a\nrotary pump and a diffusion pump, which continuously remove air from the chamber to achieve\nlow pressure conditions. In contrast, for high-pressure surface tension measurements, the\npolymer system is usually subjected to supercritical CO₂, which is introduced from the top into\na sealed high-pressure chamber to create the necessary pressure conditions, requiring a\n4\ncompletely different setup. As mentioned above, here, we primarily focus on the low-pressure\nregion; this is the range where we observed meaningful changes in surface tension. The high-\npressure region (above atmospheric pressure) was not the focus of our investigation and have\nbeen studied extensively before (See Li, H. et al. Ind. Eng. Chem. Res. 43, 509-514 (2004);\nPark, H. et al. Ind. Eng. Chem. Res. 45, 1650-1658 (2006). etc.).\nQ5. On page 5, the authors state: “A very interesting observation from this plot is the\ndescending order of 1.65, 1.51, 1.25, 1.21, and 0.41 for PEG, PS, PI, PP, and PDMS,\nrespectively.” Could the authors provide an explanation for this observed trend among the\ndifferent polymers?\nA5: Thank you for your comment. As suggested by the Reviewer, we have incorporated\nadditional experimental data (see A1 for details) and carried out a new data fitting procedure\nusing the Hill equation of adsorption.\nSpecifically, we used the following expression to fit the data,\n(𝑃/𝑃 )𝐻\n0\n𝛾 = 𝛾 +(𝛾 −𝛾 ) .\n0 ∞ 0 1+(𝑃/𝑃 )𝐻\ndescribed by this equation well for a Hill coefficient of 𝐻 = 0.20. Mathematically, the Hill\nequation describes a sigmoidal curve. In the Log-linear plot derived from Hill’s equation, the\nparameter 𝐻 describes the sharpness of the transition while 𝑃 describes the transition point.\nparameter 𝑃 is related to the air-polymer interaction strength. However, since the Hill equation\n0\nis a phenomenological model for adsorption, a rigorous derivation of the relationship between\nthe parameters and molecular model is not available.\nQ6. According to the fitting results (Table 1), the K value is 4 and appears to be invariant\nacross different polymer types. Since the Langmuir constant (K) reflects the extent of\ninteraction between the adsorbate and the surface, could the authors clarify whether the K\nvalue in Table 1 was derived from fitting or assumed? Additionally, could the authors explain\nwhy the K value remains constant across different polymers?\nadditional experimental data. These modifications enabled us to establish a universal master\ncurve that reliably captures the behaviour across the different polymers (see A1 and A5 for\ndetails). The fact that all the data fall on one master curve demonstrate that the physics\ngoverning the behaviour of the surface tension as a function of air pressure at low pressure\nregion is universal, independent of the polymer samples. The new K values for PEG, PS, PI,\nPP, and PDMS estimated from the fitting estimated by 𝑃 = 𝐾−1/𝐻, where H is assumed as 0.2\n0\nare 1.57, 0.97, 0.34, 1.24, 0.62, suggesting that the affinity and/or binding strength between a\nsurface and adsorbate is highest for PEG and lowest for PDMS. The variation is reasonable\nexpect for PI; we speculate that the unlikely K value for PI might be caused by the degradation\nproblem.\nQ7. The authors suggest that the observed pressure effect should not be specific to polymers\nand should be universally applicable to any physiochemical systems. However, they do not\nprovide any measurement data for other material types, such as ionic liquids, surfactants, or\nrigid molecules. Moreover, the lattice model and Langmuir isotherm employed appear overly\nsimplistic. Polymers, particularly block copolymers, exhibit surface inhomogeneities and non-\none-dimensional interfaces due to fluctuating chain ends, making the Langmuir adsorption\nisotherm an inadequate approximation for such systems.\nA7: We thank the Reviewer for the comments. In the revised work, we have adopted a new\ntheoretical model and fitted the experimental data, which allowed us to establish a universal\nmaster curve. This approach demonstrates that the observed pressure effect, the reduction of\nsurface tension under decreased pressure at elevated temperature, is consistently captured\nacross all polymer systems studied. Our work focuses on a general thermodynamic effect,\nwhere reduction of pressure under high temperature leads to a decrease in surface tension of\npolymer system. We agree with the Reviewer that the statement about the universality of the\nobservation needs more supports. It will be interesting to carry out the measurements for other\nmaterial types, such as ionic liquids, surfactants, or rigid molecules in the future. We also agree\nwith the comment that surface heterogeneity could affect the air adsorption. Since all polymer\nsurfaces are heterogeneous in nature, the fact that all the data points are well described by the\nHill equation with a small Hill coefficient could be related to this. Further experimental and\ntheoretical studies are desirable to resolve this point.\ntension as a function of pressure, but from what I could tell, they only used linear fits with two\ndata points per line (Fig 5). If I am in error here, and more data points are used, the plot should\nbe changed for clarity, since each line appears only to have 2-3 points over a broad pressure\nrange. Furthermore, the cohesive energy of the polymers used is known; how do the cohesive\nenergies extracted from this fit compare with literature values?\nA8: Thank you for the comment. As this is the similar comment as Q1 and we have responded\nto this comment there (see A1 for details).\nQ9. The research is of broad interest, but I think some work needs to be done to validate their\nmodel of lowering interfacial tension due to gas adsorption.\nA9: We appreciate the reviewer’s comment. We have revised the model and included\nadditional data points to strengthen its validation. With these updates, we believe the\nexplanation of the observed phase behaviors is now more robust and better supported.\nSome further comments:\nQ10. Page 5; last two sentences of 1st paragraph – “This could be presented better. The way\nthis is worded now sounds like they are saying more than just the temperature change is varying\nthe surface tension even though the pressure is being kept constant. I think they mean to say\nthat the surface tension could be changed more dramatically by changing other factors, such\nas pressure.”\nA10: As suggested by the Reviewer, last two sentences of 1st paragraph on Page 5 were revised\nas shown below and added to the revised manuscript.\n“As a result, the surface tension variation could be changed more dramatically by changing\nother factors, such as pressure.”\nQ11. Page 5; middle of 2nd paragraph – “Should use a log scale for pressure in this graph to\nbetter illustrate this point because most of the important data points are overlapping\nsignificantly and the pressure at which the decrease happens isn’t obvious.”\nPage 5; second to last sentence of 2nd paragraph – “Okay, so they do this in Figure 3 but why\neven include figure 2 then since it’s so unclear?”\nFigures 2 and 3 have been removed, and Figure R1 (Figure 2 in the revised manuscript) has\nbeen added to present the results in a log–linear plot.\nQ12. Page 5; last sentence of 2nd paragraph – “What does this conclusion even mean? There\nare no units, and the plot doesn’t clearly reflect these values.” – I see now that they probably\nmean the slope values, but they don’t draw any conclusions about the significance of this\ndecrease here or anywhere else in the manuscript.\nA12: We thank the Reviewer for pointing this out. We have removed the sentence on Page 5\n(2nd paragraph, last sentence) that explicitly refers to the slope and its qualitative significance\nbased on our new model is described as below.\nWe have revised our theoretical model and incorporated additional experimental data.\nThese modifications enabled us to establish a universal master curve that reliably captures the\nbehaviour across the different polymers (see A1 and A5 for details). The fact that all the data\nfall on one master curve demonstrates that physics governing the behavior of the surface\ntension as a function of air pressure at low pressure region is universal, independent of the\npolymer samples. The K values for PEG, PS, PI, PP, and PDMS estimated from the fitting\nestimated by 𝑃 = 𝐾−1/𝐻, where H is assumed as 0.2 are 1.57, 0.97, 0.34, 1.24, 0.62,\n0\nsuggesting that the affinity and/or binding strength between a surface and adsorbate is highest\nfor PEG and lowest for PDMS. The variation is reasonable expect for PI; we speculate that the\nunlikely K value for PI might be caused by the degradation problem.\nQ13. Page 6; middle of 1st paragraph – “Normally I’d ask what the pdi is for each polymer\nhere, but, since we’re dealing with a melt sample and the trend remains unchanged, perhaps it\nisn’t the most relevant. Depending on what’s said about the surface-air adsorption part of their\ntheoretical model, the pdi may or may not be important to consider.”\nA13: We thank the Reviewer for the comment. Note that all measurements were conducted in\nthe melt state, and the observed log-linear relationship between surface tension and pressure\nremains consistent regardless of molecular weight. In this context, the influence of\npolydispersity index (PDI) is expected to be minimal.\nQ14. Page 9; middle of 1st paragraph – “Is this for the same scenario (high-pressure) as the\nprevious sentence?”\nsentence.\nThe reduction of surface tension in the high-pressure region is primarily attributed to two\nfactors:\n(i) the increased dissolution of the gas molecule (solubility) with the increase in\npressure, and\n(ii) the reduction in density difference between the gas molecule and polymer at the\ninterface.\nThe corrected sentence: “The drop in surface tension in high pressure can also be explained in\nterms of density which is mostly attributed to the….” was added on Page 08 in the revised\nmanuscript.\nQ15. (a) Page 10; end of 1st paragraph – “Back to my earlier point about pdi, perhaps pdi is\ncontributing to this difference here since lower MWs preferentially locate at the surface and\nyou cite air-polymer interactions at the interface to be the main contribution to the surface\ntension reduction. Also, what specifically is contributing to the differences in the slope values\nfor each polymer?”\nA15: (a) Thanks for your comment. While we agree that lower molecular weight species can\npreferentially migrate to the surface, we note that the relationship between surface tension and\npressure is consistently observed across polymers with both high and low molecular weights\n(as shown in Fig. R1). This suggests that polymer chain length and, by extension, PDI may not\nplay a dominant role in the observed trend. Instead, we attribute the differences primarily to\nintrinsic polymer-air interfacial interactions rather than molecular weight distribution effects.\nQ16. (b) Also, what specifically is contributing to the differences in the slope values for each\npolymer?”\nA16: (b) As previously discussed in A5, we have already provided an explanation for this point.\nBriefly, we have clarified the factors contributing to the differences in the slope values for\ndifferent polymers.\nQ17. (c) They should also mention why they think PI is different from the rest of the polymers.\nA17: (c) As compared to other polymer systems, PI shows a different behavior in that the fitting\nparameter 𝑃 is much larger than that for all other samples. We speculate that this might be\n0\nattributed to thermal degradation of PI since its degradable temperature is very close to the\n9\nexperimental temperature (~200 °C). The degradation could lead to subtle changes in surface\ncomposition or reactivity, thus resulting in a rather different fitting parameter. This was added\non Page 09-10 in the revised manuscript.\nComment #1: The manuscript presents an interesting experimental study of the surface tension\nof polymer melts at low pressures. A threshold pressure is observed below which surface\ntension and pressure feature a strong positive correlation. To rationalize the existence of the\npressure threshold the authors introduce a Langmuir isotherm model that justifies the\nexistence of a critical pressure threshold with the hyperbolic shape of the isotherm.\nOverall this is an interesting, innovative and well-executed study. Only few experimental\nstudies of the surface tension of liquids (and polymers) have been reported in the low-pressure\nregime, thus there is potential that this study to stimulate follow-up work. It is noted that the\nperformance of materials in low-pressure environments is increasingly receiving attention due\nto novel application demands, for example, in space exploration.\nReply: Thanks for the positive comments.\nComment #2: However, while I view this study very favorably overall, I am not compelled to\nrecommend publication in Nature Communications. Although the study is pioneering in\ncharacter, the scope is rather niche. Although the results could stimulate follow-up research\nin both fundamental and application-related areas for polymer materials, the focus of such\nactivities is not elaborated in the proposal. There are also some 'loose ends' which leave the\nreader puzzled.\nReply: Thanks for the nice suggestions to strengthen the novelty and impact of this manuscript.\nThe following sentences were added to emphasize the observed phenomena. Only few\nexperimental studies of the surface tension of liquids (and polymers) have been reported in the\nlow-pressure regime, thus there is potential that this study to stimulate follow-up work. Also,\nthe finding that surface tension increases with pressure in vacuum conditions challenges\nconventional understanding derived from high-pressure studies and thus give implications for\npolymer processing and surface science. We also point out that we have revised the fitting\nmodel and found that all the data points fall on a master curve. This indicates that the observed\nbehavior for different polymer samples must be governed by the same physics. This\nobservation alone would be of great interest to the community.\nQ1: For example, what is the role of chemistry on the observed differences between the different\npolymeric systems, molecular characteristics (for example, differences in dispersity indices)\nbetween the polymers are not mentioned.\nmolecular characteristics. We agree with the reviewer that differences in dispersity indices and\nchemical structure exist among the polymer samples. On the other hand, the revised fitting\nprocedure demonstrates that all the data points can be described by a master curve. This\nobservation implies that the observed behavior is governed by the same physics, whereas the\ndetails of polymer chemistry and molecular characteristics play a less role.\nQ2: I did not find a description of the actual sample preparation process. For example, were\nmaterials solution processed or heat-pressed, where the materials, were materials conditioned\nby vacuum annealing prior to testing?\nA2: We thank the reviewer for the comment. As described in the Materials section, the polymer\nsamples, whether commercially obtained or prepared in the laboratory, were used without\nadditional processing such as solution casting or heat pressing. The samples were either in\npowder form (e.g. PEG, PS, PP) or in viscous liquid-like form (e.g. PI and PDMS), and were\nplaced directly into clean glass vials. Each sample was placed in a clean glass vial along with\nthe suspended capillary tube and kept in a homemade vacuum oven. The rotary and diffusion\npump were operated overnight to achieve a high degree of vacuum in the chamber. Following\nthis, the oven was heated to the desired temperature while maintaining vacuum conditions. This\nprocess enabled thermal annealing under high vacuum prior to testing. This description was\nadded on Page 12 in the revised manuscript.\nQ3: It would have been interesting to evaluate for one chemistry the role of chain end density,\nfor example, by variation of the molecular weight.\nA3: Thank you for the comment. As shown in Fig. R1 (Fig. 2 in the revised manuscript), we\ninvestigated the effect of molecular weight and observed a consistent relationship, irrespective\nof molecular weight. We have obtained surface tension data for a low-molecular weight PEG\nsample as well and obtained consistent behavior. This observation suggests that the observed\npressure dependence of surface tension is robust.\nComments: This manuscript investigates the influence of air pressure on the surface tension\nof polymer melts under high vacuum conditions, employing a custom-built vacuum apparatus\ncombined with the pre-coated capillary height method. The authors report an intriguing\nanomalous behavior: while surface tension 𝛾 decreases with increasing temperature as\nexpected, it increases logarithmically with air pressure over a wide range 10⁻⁴to10⁵N/m².\nThey\nfurther propose a theoretical framework based on a lattice gas model and Langmuir adsorption\nisotherm to rationalize the observed pressure dependence, attributing it to air adsorption\neffects at the polymer surface. Strengths of the study include the novel exploration of polymer\nsurface tension under high vacuum—a relatively underexplored regime—and the integration\nof experimental data with a theoretical adsorption model. The finding that surface tension\nincreases with pressure in vacuum conditions challenges conventional understanding derived\nfrom high-pressure studies and may have implications for polymer processing and surface\nscience.\nReply: Thanks for the positive comments. As suggested by the Reviewer, the questions\nmentioned were addressed (see below for details).\nHowever, several limitations currently hinder the manuscript’s impact and clarity:\nQ1. Experimental Setup Description: The homemade vacuum oven is central to the study, yet\nits design, operational principles, and advantages over conventional systems are insufficiently\ndetailed. Moreover, the integration of the vacuum environment with the surface tension\nmeasurement technique requires clearer explanation to ensure reproducibility and to validate\nthe experimental approach.\nA1: Thank you for the comment. A detailed description of the homemade vacuum oven,\nincluding its design, operational principles, and integration with the surface tension\nmeasurement technique, has already been provided in the Supplementary Information as shown\nbelow.\nTo investigate the pressure dependence of surface tension for various polymeric melts, a\nhomemade vacuum oven, as illustrated in Fig. R5A was designed and constructed, along with\nthe image of the actual setup shown in Fig. R5B and C. The oven was constructed from #304\nstainless steel and featured a heating chamber connected to a LEYBOLD S8B mechanical\npump and a TOKYO JAPAN DPF-4ZS diffusion pump, which can operate simultaneously. An\nInstruTech IGM402 Hornet pressure sensor was incorporated into the system to monitor\nback side images of the actual setup of the oven.\nvacuum conditions. The heating chamber has a working volume of 0.3×0.3×0.3 m3, providing\nsufficient space for using the capillary height method to measure the surface tension of the\nsamples. The side walls have a thickness of 10 mm to enhance the strength of the chamber\nunder high vacuum conditions. Several design considerations were made to ensure precise\ncontrol of the environment for surface tension measurements, as described below. An O-ring\nwas embedded in the chamber door to create an airtight seal for maintaining a high vacuum,\nwhile a rotary valve was installed to adjust the vacuum conditions. Moreover, a temperature\nsensor was placed in the middle of the working space to monitor the actual temperature of the\nsample, ensuring accurate and optimized automatic heating. Finally, a water-cooling pipeline\n15\nwas installed along the edge to protect the O-ring from expansion caused by high temperatures.\nThe homemade vacuum oven can achieve a maximum temperature of 400°C and reach an\nultimate vacuum level of approximately 10-4 Pa.\nSeveral methods can be employed to measure the surface tension of polymers, including\nthe capillary height method, pendent drop, rotating bubble, sessile bubble, Wilhelmy plate, and\nmaximum bubble pressure methods. Owing to the viscous nature and lower degradation\ntemperature of polymeric melts, it is essential to choose a suitable approach for accurate\nmeasurements. Note that our focus is on the dependence of surface tension in specific\nenvironments, particularly under high-vacuum conditions (⁓10⁻⁴ Pa) and at a high temperature.\nThe capillary height method was chosen due to its simplicity and reliability. The major\ndrawback of the capillary rise method is that it takes a considerable amount of time to reach\nequilibrium for polymeric materials. Furthermore, an approach involving the use of a pre-\ncoated capillary tube has been proposed to reduce friction between the tube wall and the viscous\npolymer melt, which can significantly decrease the time required to reach equilibrium. Most\nimportantly, this approach also fulfils the requirement of zero contact angle. Note that the inner\ncoating of the capillary tube has to be obtained by filling the tube through a capillary\nphenomenon followed by removing it in a vacuum oven with gravity; subsequently, by passing\na stream of gas through the tube, it is possible to create the aimed coating with smooth surface\nfor the zero-contact angle condition. The results acquired from the pre-coated capillary tube\nexperiments are well comparable to the reported surface tension at atmospheric pressure\nconditions. The equation of the capillary height method is,\n𝜌𝑔𝑟ℎ\n𝛾 = (1)\n2cosθ\n𝜌= the density of the polymer melt\ng= the gravitational acceleration\nr =the radius of the capillary tube\nh = the height of the polymer melt in the capillary tube\nθ = the contact angle of the liquid on the capillary wall.\nNote that measuring the contact angle is particularly challenging, especially in a thin capillary\ntube under the specified experimental conditions. By taking advantage of the zero-contact-\nangle condition, the measurement becomes straightforward, as the capillary rise equation can\nbe simplified to,\n𝜌𝑔𝑟ℎ\n𝛾 = (2)\n2\ncontaining the polymer sample is placed inside the homemade vacuum oven along with a\nsuspended capillary tube. The rotary and diffusion pumps are then turned on and allowed to\nrun overnight to ensure a high degree of vacuum. Once the desired vacuum level is reached,\nthe temperature is gradually increased to the target value for the surface tension measurement.\nThis description is added on Page 02-05 in Supplementary Information.\nQ2. Measurement Methodology and Validation: The pre-coated capillary height method is\nsensitive to melt density, which can vary with temperature and pressure. The authors do not\naddress whether such effects were accounted for. Additionally, validation of the measurement\nmethod against standard systems or alternative techniques is lacking, which raises concerns\nabout the robustness of the reported data.\nA2: We thank the reviewer for the comment. While melt density is temperature-dependent, its\nvariation with pressure is negligible under the high-vacuum conditions (~10⁻⁴ Pa) used in this\nstudy. In this study, the temperature-specific melt density values from the literature were used\nin our calculations.\nQ3. Contextualization with Prior Work: The manuscript references high pressure studies\nRefs.11–18 reporting linear decreases in surface tension with pressure, but does not clarify the\npressure ranges or gas environments involved. Since those studies focus on CO₂ rather than\nair, the influence of gas species on surface tension behavior should be discussed. Inclusion of\ncomplementary high-pressure experiments under similar conditions would strengthen the\ncomparative analysis.\nA3: Thanks for the comment. In this study, the homemade vacuum oven was designed using a\nrotary pump and a diffusion pump, which continuously remove air from the chamber to achieve\nlow pressure conditions. In contrast, for high-pressure surface tension measurements, the\npolymer system is subjected to supercritical CO, which is introduced from the top into a sealed\n₂\nhigh-pressure chamber to create the necessary pressure conditions, requiring a completely\ndifferent setup. As mentioned above, here, we primarily focus on the low-pressure region; this\nis the range where we observed meaningful changes in surface tension. The high-pressure\nregion (above atmospheric pressure) was not the focus of our investigation and have been\nstudied extensively before (See Li, H. et al. Ind. Eng. Chem. Res. 43, 509-514 (2004); Park, H.\n17\net al. Ind. Eng. Chem. Res. 45, 1650-1658 (2006). etc). (Reference 11-18 in the maimn\nmanuscript)\nQ4. Theoretical Model Application: While the Langmuir adsorption model is introduced, its\ncapacity to explain the temperature dependence of surface tension is not fully explored, despite\nthe temperature dependence embedded in the model parameters. Furthermore, the physical\ninterpretation of key fitting parameters and their trends across polymer systems remain unclear.\nA4: Thank you for the comment. Note 𝑃 = 𝑘𝐵𝑇 𝑒−𝛽∆𝜖 can also be described as parameter\n0 𝜆3\ndetermined by the adsorption energy and temperature. This relationship highlights that the\ntemperature dependence of surface tension is inherently captured through 𝑃 , as variations in\nwas done for the data points we had. However, most of the data points were invisible due to\nour poor choice of using a linear plot. Since the pressure change was over 8 orders of magnitude,\na linear plot compresses most of the points to a vertical line. Instead, it is the best to use Log-\nlinear plot to demonstrate the data points and fitting. We have redone the data fitting using a\nnew model, the Hill equation, and an expanded data set. Details of the fitting function are given\nbelow. All the data points and the corresponding fitting curves are shown in the new Figure R1\nthat is reproduced below. Overall, excellent fitting was obtained for all the data points over the\nrange of the pressure.\nAssuming the change of surface tension is attributed to air adsorption, we can estimate\nthe surface tension as 𝛾 = 𝛾 +𝛼𝜙, where 𝜙 is the fraction of adsorbed air molecules on the\n0\nsurface. One popular adsorption formula is the Hill’s equation,\n𝑃𝐻\n𝜙 =\n𝑃𝐻 +𝑃𝐻\n0 0\nsaturation or threshold pressure, and is a parameter indicating the turning point of adsorption.\nThe familiar Langmuir adsorption formula is recovered when 𝐻 = 1.\nThe Hill equation is used widely in biochemistry to describe the binding between\nbiomolecules, which is similar to the adsorption process as the air-polymer system. The Hill\ncoefficient is commonly found to be larger than 1 in biologically systems, indicating positive\ncorporative binding. The case with 𝐻 < 1 corresponds to negative cooperative binding. Since\nthe polymer and air molecules are not strongly interacting, a small Hill coefficient is not an\n18\nunreasonable result. The value of 𝑃 represents saturation or threshold pressure. Using this\n0\nequation, we can write the surface tension as,\n𝑃𝐻 (𝑃/𝑃 )𝐻\n0\n𝛾 = 𝛾 +𝛼 = 𝛾 +(𝛾 −𝛾 )\n0 𝑃𝐻 +𝑃𝐻 0 ∞ 0 1+(𝑃/𝑃 )𝐻\n0 0\nHere 𝛾 , 𝛾 and 𝑃 are fitting parameters when the Hill coefficient 𝐻, where 𝛾 is surface\n0 ∞ 0 0\ntension in vacuum (P=0), 𝛾 is surface tension at saturated pressure (P=∞). We have tested\n∞\ndifferent value of 𝐻 and found that a small 𝐻 value gives good fitting to all the data. After\nsome exploration, we decided to use 𝐻 = 0.20 that gives good fitting to all the data. The fitting\nwas carried out by using the Nonlinear Model Fit function of Mathematica. It is found that all\nthe data could be described by this equation with 𝐻 = 0.20 and the following fitting\nparameters are shown in Table R1 (Table 1 in the revised manuscript). An example of the data\nfitting is shown in Fig. R2 (Fig. 2 in the revised manuscript). for the case of polystyrene. The\nfitting allows us to display all the data on one plot, if we scale the surface tension using the\nfollowing scaled form,\n𝛾 −𝛾 (𝑃/𝑃 )𝐻\n0 0\n=\n𝛾 −𝛾 1+(𝑃/𝑃 )𝐻\n∞ 0 0\nfitting was carried out by using the Nonlinear Model Fit function of Mathematica. It is found\nthat all the data could be described by this equation with 𝐻 = 0.20 and the following fitting\nparameters are shown in Table R1. An example of the data fitting is shown in Fig. R2. for the\ncase of polystyrene. The fitting allows us to display all the data on one plot, if we scale the\nsurface tension using the following scaled form,\n𝛾 −𝛾 (𝑃/𝑃 )𝐻\n0 0\n=\n𝛾 −𝛾 1+(𝑃/𝑃 )𝐻\n∞ 0 0\nThe fact that all the data fall on one master curve demonstrate that physics governing\nthe behavior of the surface tension as a function of air pressure at low pressure region should be\nuniversal, independent of the polymer samples.\nTable R1. Fitted parameters for all data points.\n𝛾 𝛾 𝑃\n0 ∞ 0\nPEG 16.2 35.2 0.10\nPEGLmw 13.5 31.9 0.03\nPS 11.6 27.5 1.15\nPI 13.5 28.1 226.2\nPDMSLmw 7.6 13.4 1.19E-3\nPP 9.8 23.0 0.311\nQ5. Data Presentation and Discussion: Some figures appear redundant e.g., Fig.3 vs.Fig.2 ,\nand others e.g., Fig.5 would benefit from alternative plotting formats log−linear to better\nreveal trends. The discussion of certain results, such as those in Fig. 4, is cursory and does not\nleverage the theoretical framework to provide deeper insight.\nA5: Thanks for the comment. We agree that some figures appeared redundant and therefore\nsummarized the data showing the variation of surface tension as a function of pressure in Table\nR2 (Table 1 in the revised manuscript). Based on these data, we have replotted the results in a\nlog–linear format, which more clearly reveals the observed trends (as shown in Figure R1\n(Figure 2 in the revised manuscript)). We have also included a scaled master curve plot (as\nshown in Figure R3 (Figure 3 in the revised manuscript) to clearly compare all datasets and\nhighlight the overall trend. We have also included both high and low molecular weight data in\nthe fitting to provide a more comprehensive comparison. We believe these revisions improve\nthe clarity and consistency of data presentation.\npressures at 200°C.\nPressure (N/m2) Surface Tension 𝛾, (J/m2×103)\nPolydimethylsiloxane 100000 12.7\n(PDMS)\n316 11.4\nhigh molecular\n0.3 10.8\nweight\n4E-4 9.5\nPolydimethylsiloxane 100000 13.5\n(PDMS)\n158.49 12.6\nlow molecular\n1 12.1\nweight\n0.01 11.1\n1.00E-03 10.7\n1.00E-04 9.6\nPolypropylene (PP) 100000 22.9\n17500 21.4\n316 19.4\n0.4 17.3\n3E-4 12.3\nPolyisoprene (PI) 100000 24.9\n316 20.5\n1 17.8\n8E-4 14.3\nPolystyrene (PS) 100000 27.4\n17500 24.6\n1000 23.2\n0.1 18.2\n1E-3 15.1\n1E-04 13.1\nPolyethylene Glycol 100000 34.1\n(PEG)\n316 31.9\nhigh molecular\n0.2 26.4\nweight\n4E-4 20.9\nPolyethylene Glycol 100000 31.9\n(PEG)\n17500 29.7\nlow molecular weight\n316 28.6\n0.6 25.3\n21\n7E-4 19.4\nFig. R3. Fitting of surface tension versus pressure plot for PEG, PS, PI, PP, PDMS, and PEG\nLMW using the proposed model to obtain master curve.\nComments: In summary, this work addresses an interesting and novel topic with potential\nsignificance. However, the manuscript requires substantial revision to enhance experimental\ntransparency, methodological rigor, theoretical depth, and data presentation. I recommend\nthat the authors undertake a major revision addressing the points above before the manuscript\ncan be considered for publication in Nature Communications.\nReply: As suggested by the Reviewer, substantial revision to enhance experimental\ntransparency, methodological rigor, theoretical depth, and data presentation was conducted.\nReviewer #1\nComments: The authors have made an interesting observation that the apparent\ninterfacial tension decreases with decreasing pressure and that the reduction can be\nmapped on to a function put for by Hill to describe the behavior of receptors in\nbiological systems. Here, the authors speculate that the surface of the molten polymers\nadsorb air molecules, and it is those adsorbed molecules that give rise to the observed\nvariation in the apparent interfacial tension.\nReply: We thank the Reviewer for this succinct and accurate summary of our work.\nQ1: Unlike biological systems where receptors have specific interactions, any\ninteractions with the air molecules are left to the reader to decide. Air is a mixture of\ngases (78% N2, 21% O2 and 0.93% Ar). The authors make no attempt to discern\nbetween these and treat air as a single component gas, which it is not, nor do the\nauthors perform any control experiments to differentiate between them. Certainly, there\nmust be a difference between the adsorption of these different gases.\nA1: We thank the Reviewer for this insightful comment. Indeed, air is a homogeneous\nmixture of N , O and Ar, and different gas molecules would interact with the polymer\n2 2\ndifferently. In our simple model, the contribution of different gas-polymer interactions\nis represented by an averaged interaction parameter. Certainly, the model could be\nimproved by performing control experiments to differentiate the interaction between\nindividual molecule and polymer. However, the behaviors with respect to the pressure\ndependence of surface tension for polymer melts under vacuum should be the same\nbecause the average-interaction approximation capture the main physics mechanism\nhere. An evidence of this statement is found in the work entitled “Thin Liquid Polymer\nFilms Rupture via Defects” by Stephan Herminghaus and coworkers (see Langmuir\n1998, 14, 965-969), where they also observed that the pressure effect on polymer thin\nfilm rupture is insensitive to the gas molecules. To make this point more transparent,\nwe have added a description and discussion of the nature of our simple model on page\n10-11 in the revised manuscript.\nQ2: The master curve that the authors describe simply says that the functional form of\nthe reduction is the same, suggesting, though not proving, that origin and mechanism\nof the adsorption is similar. Normalization of the data at the 50% change point, P0,\nand the range in the change of ensures that the data will overlap. So, I am less\nimpressed with the master curve and am still not informed as to the origin of the\nobservation. Is it adsorption? Perhaps, but I would have expected some dependence on\nthe chemical composition or the density of the polymer (free volume or concentration\n1\nof chain ends at the surface or chain orientation at the surface or the interfacial tension\nof the specific polymer). None of these appear to be the case. So, what is it?\nA2: We thank the Reviewer for this thoughtful comment. The question about the origin\nof our experimental observation is a great one. Our proposed model provides a plausible\nanswer to this question, and we believe our simple model correctly captures the\nexperiments and is technically correct. That said, we agree with the Reviewer that many\nfactors, such as those mentioned by the Reviewer, could affect the surface tension of\npolymers. Our hypothesis is that the change of surface tension in the low-pressure\nregion must be related to air-polymer interaction and air adsorption provides the\nsimplest and intuitively attractive mechanism of this effect. Despite the fact that many\ndetails of the air-polymer interactions are ignored, the adsorption model with Hill\nequation does provide an adequate description of the experiments. We have added some\ndiscussion on the hypothesis on Page 10-11 in the revised manuscript.\nQ3: The adsorption of interest is only at the surface, at least that is what will change\nthe interfacial tension, but the surfaces of the different polymers are chemically\ndifferent, have densities that are different and dynamics that will be different, but the\nmeasurements done here are insensitive to these differences. As I said in my earlier\nreview, “I think some work needs to be done to validate their model of lowering\ninterfacial tension due to gas adsorption.”, and I still feel that is the case.\nA3: We thank the Reviewer for this thoughtful comment, and we agree with the\nstatement that many factors could affect the surface tension of polymers. Our aim in\nthe modeling effort is to construct a simple model that correctly captures the\nexperiments and is technically correct. Based on the good agreement between model\nprediction and experiments, we believe this goal has been achieved.\nQ4: This work originated in the research that Ho had done on the orientation of block\ncopolymer domains and there I could understand a preferential adsorption of any of a\nvariety of vapors (gases, water vapor, or other gas) that could mediate the surface\nenergies of the components. Yet, given the arguments the authors present in this\nmanuscript, I am not certain that holds. Consequently, even though I find the\nobservations interesting, I do not feel that this manuscript has the scientific rigor for\npublication in Nature Comm. I strongly encourage the authors to pursue this further,\nsince there will be some interesting physics underpinning the observed behavior, but I\njust do not see that in this manuscript.\nA4: We thank the Reviewer for this comment. We must respectfully disagree with the\nReviewer’s arguments here. Indeed, the current work is motivated by our earlier\nobservation on the effect of air pressure on the orientation of block copolymer\nnanostructures, indicating there must be a change of polymer surface tension under high\n2\nvacuum. To explore this effect, we designed the needed equipment and carried out the\nexperiments reported in the current manuscript. The key message for our current study\nis that decreasing air pressure to below 1mbar resulted in a sharp decrease of surface\ntension of different polymers. This experimental observation is an interesting and new\nresult alone. To explain the experiments, we constructed a simple model based on air\nadsorption and showed that the model provides adequate explanation of the experiments.\nTaken together, we believe the current manuscript provides a rigorous study on the\neffect of high vacuum on polymer surface tension. Publication of this work will draw\nthe attention of readers from physics, chemistry and materials community and motivate\nfurther studies on the surface tension of a wide class of materials under vacuum.\nQ5: As an aside, I do want to point out that S. Herminghaus and coworkers performed\ndome detailed studies on polymer thin films stability, dewetting, etc. where they also\nobserved difference when the homopolymers and mixtures showed different behavior\nunder vacuum or different gas atmospheres which they attributed to variation in the\ninterfacial tensions and it would be worthwhile for the authors to look at his work.\nA5: We thank the Reviewer for bringing the work of Prof. Herminghaus to our attention.\nWe did a literature survey of works by Prof. Herminghaus and found one relevant paper\npublished in Langmuir in 1998 (Jacobs, Herminghaus, and Mecke “Thin Liquid\nPolymer Films Rupture via Defects” Langmuir 1998, 14, 965-969.). The particularly\nrelevant results reported in this work are their observations of the effect of gas pressure\non thin film rupture behavior that is summarized as: “In order to investigate whether\ndefects of this type are involved in the rupture process, we repeated our studies at\ndifferent ambient pressures. Experiments were carried out in air, nitrogen, argon, and\nhelium atmospheres. A number of identically prepared samples were sealed in glass\nampules, which were evacuated before sealing to a desired pressure between\natmospheric pressure and 10-5 Pa. Afterward, the ampules were heated to 130 °C. In a\nsecond series of experiments, samples were placed directly into a vacuum oven, the\npressure of which could be varied in a wide range, and annealed up to a maximum\ntemperature of 200°C. It was generally found that when the experiments were\nperformed under vacuum below a pressure of about 1 mbar, rupture could be\ncompletely suppressed, provided the substrate had been properly cleaned. Irrespective\nof the gas used, a pressure of 10 mbar or more safely induced the formation of holes,\nwhile a pressure of 1 mbar or less reliably suppressed any rupture.”. Because thin film\nrupture is largely driven by the surface tension of the polymers, these observations\nimply a sharp decrease of polymer surface tension when the gas pressure is decreased\nto below 1 mbar, and this decrease is independent of the gas molecules. It is interesting\nto note that the observations reported in this earlier work are consistent with our results\n3\non the pressure-dependence of polymer surface tension. We have added a discussion\non this point and a new reference of the paper on Page 10-11 in the revised manuscript.\nReviewer #2\nComments: In my original review, I noted several points that needed revision prior to\npublication of the manuscript. The authors have expanded significant efforts to improve\nthe quality of the original submission and I find that the revised manuscript\ncomprehensively addresses all the points I raised (with the exception of principle\nsuitability for Nature Commun). I commend the authors for significant extension of the\ndata range, in particular, the study of M-dependence. I also appreciate the extended\ndiscussion of sample preparation procedures. While I remain skeptical about the\nsuitability of such a narrow theme for a broad audience journal, the principal quality\nis now (in my opinion) on par with work that would generally be considered suitable\nfor high impact broad audience journals. I have no further changes to suggest.\nReply: We thank the Reviewer for their positive comments.\nComments: Recommendation: Minor Revision, followed by Acceptance The authors\nhave made substantial revisions to the original manuscript, introducing the Hill\nequation to describe the anomalous surface tension behavior of various polymer melts\nunder low pressure. This work presents an interesting finding with potential\nimplications for polymer processing and surface science. The authors have addressed\nmany of the previous concerns with thoughtful responses and revisions.\nReply: We thank the Reviewer for their positive comments.\nQ1: However, several remaining points require clarification and refinement to enhance\nthe manuscript’s clarity, rigor, and accessibility. Addressing these issues thoroughly\nwill strengthen the work for publication. (1) Fig. 2 Data Fitting and Sample\nIdentification: In their response to Reviewer 1 (see Q1), the authors note that PI and\nPDMS data “appear to deviate slightly from the general trend”. However, in Fig. 2 (or\nFig. R1), PI (red) and PDMS (brown) closely follow the theoretical fitting, while PS\n(green) and PP (purple) show visible deviations. This discrepancy raises questions: A)\nCould there be an error in the legend, e.g., mislabeling of samples? B) Assuming minor\ndeviations for PI and PDMS: b1) Is PI’s deviation attributed to degradation, as\ndiscussed in the response to Q6, Q12, Q17, and manuscript (Page 9 Lines 190–195)?\nb2) What is the proposed cause of PDMS’s deviation?\nA1: We thank the reviewer for this constructive comment. Indeed, Figure 2 shows that\nall data points follow the proposed Hill function fitting, as noticed by the Reviewer.\nThe deviation of PI and PDMS data refers to the different trends in the fitting\nparameters, particularly for PI shown in red in Figure 2. The legend of Figure 2 is\ncorrect and there is no mislabeling of samples. Our explanation of the deviations is as\nfollows. For the PI samples, the deviation could be attributed to polymer degradation\nat the high temperature of 200 °C. For the PDMS samples, the deviation could be\nattributed to the small range of surface tension values leading to inaccurate fitting.\nWe have added a statement about this point in the manuscript.\nQ2: Integration of Reviewer Responses into the Manuscript: Key explanations provided\nin the response letter, e.g., regarding molecular weight (PDI and chain-end density\neffects), should be more explicitly integrated into the main text. For instance, the\ndiscussion on molecular weight in the manuscript (Pages 6, Lines 107–120) indirectly\naddresses Reviewer 1 (see Q13 and Q15) and Reviewer 2 (see Q3) but lacks clear\nconnections to these concerns. Readers may benefit from a concise summary of these\npoints to avoid ambiguity.\nA2: We thank the reviewer for the comments. As suggested by the Reviewer, the\ndiscussion has been integrated into the main text on page 6 in the revised manuscript.\n2\n“The observed pressure dependence of surface tension is consistent across polymers of\ndifferent molecular weights, indicating that polymer chain length or PDI does not play\na dominant role. Instead, the trend primarily reflects intrinsic polymer-air interfacial\ninteractions, suggesting that the pressure dependence should be universal to polymers.”\nQ3: Theoretical Model and Parameter Clarity: The derivation from Equation 4 to 7 is\nabrupt, and the physical meaning of parameter 𝛼 is not defined. The term 𝛾∞ is\ndescribed as “surface tension at infinite pressure P = ∞”, but the model is not validated\nfor high pressures. This terminology may be misleading; consider revising it, e.g.,\n“high−pressure limit surface tension” for accuracy. The fitting curves in Fig. 2 should\nnot extend to 1010Pa, as this implies unsubstantiated high-pressure predictive power.\nA3: We thank the Reviewer for this constructive comment. In our model, the parameter\n𝛾 represents the surface tension where air adsorption is effectively saturated.\n∞\nPhysically, this value may not be attainable because other changes would intervene\nwhen the pressure is high enough. This point was added to page 9 in the revised\nmanuscript.\nQ4: Consistency in Interpreting Pressure Effects: The introduction states that surface\ntension decreases at high pressure due to “gas dissolution” or “reduced density\ndifference across the interface”. The lattice model attributes low-pressure behavior to\nchanges in “nearest-neighbor polymer-air interactions”. Are these explanations for low-\nand high-pressure regimes consistent, or does gas dissolution also manifest as a change\nin nearest-neighbor interactions? This should be clarified to avoid perceived\ncontradictions.\nA4: We thank the reviewer for this observation. The low- and high-pressure regimes\ninvolve distinct mechanisms. At low pressure, surface tension changes are primarily\ngoverned by modifications in polymer–air interactions, as captured by the lattice model.\nAt high pressure, the decrease in surface tension arises from gas dissolution or the\nreduced density difference across the interface. We will clarify this distinction in the\nmanuscript to avoid any perceived inconsistency. This point was added to page 5-6 in\nthe revised manuscript\nQ5: Temperature Effects in the Hill Equation Model: The authors mention that the Hill\nequation can incorporate temperature response to Reviewer 3 (see Q4). Does the\ncurrent model explain the temperature dependence of surface tension observed in Fig.\n1? If so, this should be explicitly demonstrated or discussed; if not, limitations should\nbe noted.\ndecrease of surface tension as the temperature is increased. When the Hill function is\nused, the temperature-dependence of the surface tension is embedded in the parameter\n1 𝐻\n𝑃 in our model. ssing the approximate expression 𝜙 ≈ (1+ ln𝑃/𝑃 ) ,\nthe\n0 0\n2 2\ntemperature-dependence shown in Figure 1 could be explained. We have added a\ndescription of this discussion on Page 9 in the revised manuscript.\nMinor Revisions:\nQ6(1): Page 5 Line 99: “drastic reduction in surface tension … below atmospheric levels\n105Pa” is inconsistent with Fig. 2, where the reduction begins ~ 103Pa. Revise this\nstatement to reflect the data. Standardize pressure units to Pa or N/m² throughout\nthe text, figures, and captions for consistency.\nA6(1): We thank the Reviewer for the comment. The standardized pressure unit is\nchanged to N/m2 in the revised manuscript.\nQ6(2): Provide a reference for the Hill equation to contextualize its application.\nA6(2): We thank the Reviewer for the suggestion; as suggested, the reference was added.\nQ6(3): Page 9 Line 189: “the observed surface tension data could be described by a\nlinear function of lnP, as shown in Fig. 3.” However, Fig. 3 does not clearly\ndemonstrate linearity in lnP. Either revise the text or adjust the figure to highlight this\nrelationship.\nA6(3): We thank the Reviewer for the comment. The change has been added to page 9\nin the revised manuscript.\nReviewer #2 (Remarks to the Author):\n[Note from the Editor: Reviewer #2 was asked to assess the response to reviewer #1.]\nComments: This is my third review of this manuscript. Details of my assessment can be found in\nmy prior reviews, here I will focus only on my summative assessment based on the revisions that\nthe authors provided with respect to mine and other referees.\nWith respect to my comments, I concluded that this work is interesting and of high quality. Since\nthe authors pioneer a parameter range that has not been addressed in the literature, this work\nmight instigate follow-up research and receive attention by the community. However, I also noted\nthat the work is rather specialized and its reach likely narrow which, in my opinion, renders the\nwork inherently suitable for a specialized research journal such as Langmuir.\nWith respect to the comments of other reviewers, I find that the authors have provided constructive\ncomments. Comments of Referee 3 have, in my opinion, been comprehensively addressed. However,\nwith respect to reviewer 1, I find that the response has been partial. Authors acknowledge that the\npoints raised by the referee are valid, however, point to the pioneering character of the work to\njustify that the questions raised by the reviewer would largely be left for future work and perhaps\nfor different groups.\nI understand the author's bid that the manuscript should be published in its present form. The\nexperiments are novel and creative and point to potentially new physics involving polymer\ninterphases. At this stage, a semi-quantitative model that captures data is, in principle, appropriate\nas it provides a basis for future research. However, the questions raised by reviews 2 stand and\nunless addressed by the authors, cast doubt on whether significant progress has truly been\nachieved or if results are perhaps misinterpreted due to the rather limited scope of the study. I\ncontinue to believe that the work will be better served by publication in a specialized journal such\nas Langmuir.\nReply: We thank the Reviewer for their positive comments. We are pleased that the reviewer\nfound that “the authors pioneer a parameter range that has not been addressed in the literature,…”\nand “At this stage, a semi-quantitative model that captures data is, in principle, appropriate …”.\nSurface tension is a property of all materials and understanding the pressure dependence of surface\ntension in high vacuum is rarely studied. The universal behavior reported in the current manuscript\nimplies that the observed pressure dependence is likely applicable to materials beyond polymers.\nFrom this perspective, the current work would be of interest to a wider range of readers.\nComments: The authors have addressed all concerns with detailed responses.\nReply: We thank the Reviewer for their positive comments. We are pleased that the reviewer\nfound our response to the reviewer’s comments are satisfactory.\nComments: This paper reports observation of an S-shaped pressure dependence of surface tension\nin several polymers, and the development of a model based on the Hill equation that explains the\ndata.\nThe investigation is quite interesting, with findings that could have both fundamental and practical\nimplications for polymer applications.\nReply: We thank the Reviewer for their positive comments.\nHowever, I have several questions regarding the research, experiments, and theory that I would\nlike the authors to address. If these questions are adequately answered, I would support the\npublication of the paper.\nQ1: It will enhance the introduction if the authors provide a referenced overview for the pressure\ndependence of surface tension in polymers and the essential physical factors affecting this\ndependence.\nA1: We thank the reviewer for this helpful comment. We have added the following overview of\nthe key mechanisms governing the pressure dependence of surface tension in polymer systems on\npage 3 of the manuscript.\n“The pressure dependence of 𝛾 , determined under high-pressure conditions, was observed to\ndecrease linearly when the pressure is increased at a constant temperature11-18. The decrease of 𝛾\nwith increasing 𝑃 is attributed to either increased gas dissolution or the reduction of density\ndifference between the two phases across the interface11-18. However, the pressure dependence of\n𝛾 under high-vacuum conditions has not been studied prior to the current work.”\nQ2: The surface tension values in Supplementary Tables 1 and 2 do not fully agree. Could it be\ncaused by the data in Supplementary Table 1 being taken at different temperatures?\nA2: We thank the Reviewer for this helpful comment. We can confirm that the apparent\ndiscrepancy arises from the fact that the data in Supplementary Table 1 were collected from the\nliterature at room temperature and atmospheric pressure, whereas the data in Supplementary Table\n4\n2 were experimentally determined in this work at atmospheric pressure over a temperature range\nof 170~220 °C.\nQ3: Please provide information on the radius of the capillary and how it was cleaned.\nA3: We thank the reviewer for this comment. A glass capillary with an inner diameter of 1 mm or\nan inner radius of 0.5 mm was used for the surface tension measurements. Prior to each\nmeasurement, the capillary was cleaned with acetone and ethanol, and then sprayed with\ncompressed air to remove any residual contaminants. We have added this on Page 3 in the revised\nSupplementary Information.\nQ4. Provide the molecular weight and molecular weight distribution of the polymers as well as\ntheir source.\nA4: Details of the samples are already listed in the Materials and Methods (Materials) section on\npages 11–12, as shown below.\n“The polyethylene glycol (Mw = 20,000 g/mol, catalogue no. A17925.30) from Thermo Scientific\nand polyethylene glycol (Mw = 10,000 g/mol, product no. P6667) from Sigma-Aldrich used in this\nstudy are commercially available products. Polystyrene (Mw = 20,000 g/mol, PDI = 1.06),\npolyisoprene (Mw = 19,000 g/mol, PDI = 1.03), and polydimethylsiloxane (Mw = 21,000 and\n38,000 g/mol, PDI = 1.09) were synthesized in the laboratory. Polypropylene (Mw = 12,000 g/mol,\nproduct no. 428116) used is a commercially available product and obtained from Sigma-Aldrich.”\nQ5: What is the established molecular weight dependence of polymer surface tension?\nComparing\nthis with your experimental observation would be beneficial.\nA5: We thank the reviewer for this insightful comment. While the molecular weight dependence\nof polymer surface tension has been reported under ambient conditions Generally, surface tension\nincreases with molecular weight, as observed for PEG in our study. For PDMS, however, no clear\nincrease is seen, likely because its surface energy is very low, so any molecular weight–dependent\nchange is within the experimental error. However, to our knowledge, such data is not available\nunder the low-pressure conditions studied in our experiments, so a direct comparison with\nliterature values is not possible.\nQ6: I believe that the 𝜙 in equation (2) is the concentration of air over the polymer surface, while\nthe 𝜙 leading to equation (5) is the concentration of air over the whole vacuum chamber. These\n5\ntwo concentrations need not correlate. For example, if the enthalpy of adsorption is very large,\nthe adsorbed air molecules are expected to fully cover the polymer surface irrespective of the air\nconcentration in the chamber.\nA6: We thank the Reviewer for, and fully agree with, this insightful comment. The air\nconcentration 𝜙 appearing in Eq. 2-4 refers to the air concentration near the surface, which would\nbe different from the air concentration in the whole vacuum chamber when air adsorption occurs.\nEq. 5 is derived by assuming no air adsorption, whereas Eq. 6 (Hill function) is obtained by\nassuming that the air adsorption follows a sigmoidal function. This Hill function (Eq. 6) does have\nthe required property of fully covering the surface at high pressure or very small 𝑃 correspoding\n0\nto large adsorption enthalpy.\nQ7: Equation (5) suggests that the temperature dependence of surface tension has a 1/T term,\nwhich does not appear to align with the linear dependence observed in this experiment (Figure 1).\nCan the authors reconcile this apparent discrepancy?\nA7: We thank the Reviewer for this insightful comment. Qualitatively, Eq. (5) does predict a\ndecrease of surface tension as the temperature is increased. However, this expression was obtained\nby assuming that no air adsorption has occurred. As such, this expression could only be\nqualitatively correct. The data shown in Figure 1 were obtained at atmospheric pressure, which is\nin line with the literature where a linear decrease of the surface tension is reported. Currently, there\nis no theoretical model predicting this linear T-dependence. It is generally believed that this linear\nbehavior is largely due to the small temperature range of the measurements.\nQ8: The fitted values of P vary across all polymers, not just PI. Can the authors explain this\n0\nvariation?\nA8: We thank the reviewer for this helpful comment. The parameter 𝑃 is related to air-polymer\n0\ninteraction and temperature. The variation of fitted values of 𝑃 reflects the different air-polymer\n0\ninteractions ∆𝜀 . In the case of aangmuir adsorption isothermal, 𝑃 ∝ 𝑒−∆𝜀/𝑘𝐵𝑇 . However, no\n0\nexplicit expression of 𝑃 is available for the general case of 𝐻 ≠ 1 . Experimentally, the fitted\n0\nparameter 𝑃 varies across all polymers, except the PI sample with a notably abnormally large\n0\nvalue (𝑃 ≈ 226 ), the 𝑃 values of all other polymers range from approximately 0.001 to 9.6,\n0 0\nreflecting the variation of air affinity to the different polymers.\nfamiliar Langmuir adsorption isotherm and H < 1 corresponds to negative cooperative binding.\nIt would be interesting to cite systems with a H value comparable to 0.2.\nA9: We thank the Reviewer for this helpful suggestion. Negative cooperative binding is generally\nobserved in ligand-protein binding. One relevant reference on this topic is: H. Abeliovich. “An\nEmpirical Extremum Principle for the Hill Coefficient in aigand-Protein Interactions showing\nNegative Cooperativity”, Biophysical Journal, 89, 76-79 (2005). The following description and\nreference are added on Page 9 in the revised manuscript.\n“Such negative cooperative binding (H=0.2) is generally observed in ligand-protein binding.”\nQ10: I do not believe that Ref. 34 is a suitable experiment for comparison with this work. In that\nexperiment, the goal is to show that vacuum removes debris and so reduces the probability of\ndewetting. In this work, vacuum lowers surface tension, which can conversely increase the rate of\ndewetting if holes nucleate.\nA10: We thank the Reviewer for this constructive comment. Indeed, the main objective of this\nreference is the study of debris remove under vacuum conditions However, one observation from\nthis study is quite relevant to the work reported in the current manuscript. Specifically,\nHerminghaus et al. (Ref. 34) observed that, when experiments were performed under vacuum\nbelow about 1 mbar, rupture of their polymer thin film samples could be completely suppressed\nprovided the substrate had been properly cleaned. Because thin film rupture is largely driven by\nthe surface tension of the polymer, these observations imply a large decrease of polymer surface\ntension when the pressure is reduced below 1 mbar. These findings, although not quantitative, are\nconsistent with our results on the pressure dependence of polymer surface tension. This reference\nis included in our manuscript based on the above reasonings.\nSeveral minor suggestions:\n11(a): Consider revising page 6 lines 6-7 to “Also, low MW (21,000 g/mol) PDMS is referred to\nas PDMS LMW and high MW (38,000 g/mol) PDMS is referred to as PDMS?”\nA11(a): We thank the Reviewer for the suggestion. As suggested by the Reviewer, the line was\nrevised in the revised manuscript.\n11(b): Both the units and measurement uncertainties for the data in Table 2 are missing.\nA11(b): We thank the Reviewer for the comment. As suggested by the Reviewer, the units and\n7\nmeasurement uncertainties for the data in Table 2 were added on page 18 in the revised manuscript.\nThese parameters were obtained via best fit using Eq. 7 and we have not tried to estimate their\nuncertainties given the qualitative nature of the fitting formula.\n11(c): Measurement uncertainties for the data in Supplementary Table 1 are also missing.\nA11(c): We thank the reviewer for this comment. The values shown in Supplementary Table 1 are\ntaken from the literature, where uncertainties were not reported. We chose these five commonly\nused polymers (PDMS, PP, PI, PS, and PEG) for our study to illustrate that they exhibit an\nincreasing trend in surface tension at room temperature, and the table was included primarily to\nhighlight this trend.\n8"}, {"round": 3, "reviewer_comments": {"reviewer_2": "[Note from the Editor: Reviewer #2 was asked to assess the response to reviewer #1.]\nThis is my third review of this manuscript. Details of my assessment can be found in my prior reviews, here I will focus only on my summative assessment based on the revisions that the authors provided with respect to mine and other referees.\nWith respect to my comments, I concluded that this work is interesting and of high quality. Since the authors pioneer a parameter range that has not been addressed in the literature, this work might instigate follow-up research and receive attention by the community. However, I also noted that the work is rather specialized and its reach likely narrow which, in my opinion, renders the work inherently suitable for a specialized research journal such as Langmuir.\nWith respect to the comments of other reviewers, I find that the authors have provided constructive comments. Comments of Referee 3 have, in my opinion, been comprehensively addressed. However, with respect to reviewer 1, I find that the response has been partial. Authors acknowledge that the points raised by the referee are valid, however, point to the pioneering character of the work to justify that the questions raised by the reviewer would largely be left for future work and perhaps for different groups.\nI understand the author's bid that the manuscript should be published in its present form. The experiments are novel and creative and point to potentially new physics involving polymer interphases. At this stage, a semi-quantitative model that captures data is, in principle, appropriate as it provides a basis for future research. However, the questions raised by reviews 2 stand and unless addressed by the authors, cast doubt on whether significant progress has truly been achieved or if results are perhaps misinterpreted due to the rather limited scope of the study.\nI continue to believe that the work will be better served by publication in a specialized journal such as Langmuir.", "reviewer_3": "The authors have addressed all concerns with detailed responses.", "reviewer_4": "This paper reports observation of an S-shaped pressure dependence of surface tension in several polymers, and the development of a model based on the Hill equation that explains the data.\nThe investigation is quite interesting, with findings that could have both fundamental and practical implications for polymer applications.\nHowever, I have several questions regarding the research, experiments, and theory that I would like the authors to address. If these questions are adequately answered, I would support the publication of the paper.\n1. It will enhance the introduction if the authors provide a referenced overview for the pressure dependence of surface tension in polymers and the essential physical factors affecting this dependence.\n2. The surface tension values in Supplementary Tables 1 and 2 do not fully agree. Could it be caused by the data in Supplementary Table 1 being taken at different temperatures?\n3. Please provide information on the radius of the capillary and how it was cleaned.\n4. Provide the molecular weight and molecular weight distribution of the polymers as well as their source.\n5. What is the established molecular weight dependence of polymer surface tension? Comparing this with your experimental observation would be beneficial.\n6. I believe that the φ in equation (2) is the concentration of air over the polymer surface, while the φ leading to equation (5) is the concentration of air over the whole vacuum chamber. These two concentrations need not correlate. For example, if the enthalpy of adsorption is very large, the adsorbed air molecules are expected to fully cover the polymer surface irrespective of the air concentration in the chamber.\n7. Equation (5) suggests that the temperature dependence of surface tension has a 1/T term, which does not appear to align with the linear dependence observed in this experiment (Figure 1). Can the authors reconcile this apparent discrepancy?\n8. The fitted values of P vary across all polymers, not just PI. Can the authors explain this variation?\n0\n9. The authors found that the value of H is 0.2. They point out that H = 1 corresponds to the familiar Langmuir adsorption isotherm and H < 1 corresponds to negative cooperative binding. It would be interesting to cite systems with a H value comparable to 0.2.\n10. I do not believe that Ref. 34 is a suitable experiment for comparison with this work. In that experiment, the goal is to show that vacuum removes debris and so reduces the probability of dewetting. In this work, vacuum lowers surface tension, which can conversely increase the rate of dewetting if holes nucleate.\n11. Several minor suggestions:\na. Consider revising page 6 lines 6-7 to “Also, low MW (21,000 g/mol) PDMS is referred to as PDMS LMW and high MW (38,000 g/mol) PDMS is referred to as PDMS?”\nb. Both the units and measurement uncertainties for the data in Table 2 are missing.\nc. Measurement uncertainties for the data in Supplementary Table 1 are also missing."}, "author_response": ""}]

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