Introduction

The surface tension (\(\gamma\)) 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 under ambient pressure of the order of 105 Pa5,6,7,8,9,10. Similarly, the pressure dependence of \(\gamma\), determined under high-pressure conditions, was observed to decrease linearly when the pressure is increased at a constant temperature2,11,12,13,14,15,16,17. The decrease of \(\gamma\) with increasing \(P\) is attributed to either increased gas dissolution or the reduction of density difference between the two phases across the interface2,11,12,13,14,15,16,17. 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 conditions, 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 nanostructure18. 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 vacuum19,20. 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 Fig. 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 cases21,22,23. However, their performance and sensitivity significantly decrease when the drop shape becomes nearly spherical24,25,26,27,28,29,30. In the current study, the pre-coated capillary technique was chosen due to its simplicity and reliability19. 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 conditions, 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 the 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 work19. The data for PI was not available for temperatures over 200 °C due to thermal degradation of the sample. In agreement with the literature5,6,7,8,9,10, 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.

Fig. 1: Temperature dependence of surface tension of polymer melts.
Fig. 1: Temperature dependence of surface tension of polymer melts.The alternative text for this image may have been generated using AI.
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Surface tension (γ) as a function of temperature for different polymers (PEG (green filled circle), PS (red filled circle), PI (purple filled circle), PP (black filled circle), and PDMS (blue filled circle)) under ambient pressure ( ~ 105 N/m2). In this study, symbols represent experimental data points, and dashed lines indicate linear fits to the experimental data. For all polymers, γ decreases monotonically with increasing temperature. Error bars represent the standard error of the mean from five independent measurements.

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 is 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 best to use a log-linear plot to demonstrate the change in 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 \(\gamma\) is observed when the pressure is increased2,11,12,13,14,15,16,17. 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 described below. At high pressure, the decrease in surface tension arises from gas dissolution or the reduced density difference across the interface.

Fig. 2: Pressure dependence of surface tension of polymer melts.
Fig. 2: Pressure dependence of surface tension of polymer melts.The alternative text for this image may have been generated using AI.
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Log-linear plot of surface tension data for PEG (blue filled circle), low molecular weight PEG (PEGLmw) (orange filled circle), PS (green filled circle), PI (red filled circle), PP (purple filled circle), PDMS (red filled circle), and low molecular weight PDMS (PDMSLmw) (light blue filled circle) in the high vacuum range ( ~ 105−10–4 N/m2) at 200 °C. The measured surface tension decreases systematically with decreasing air pressure, with the trend following the Hill equation. Symbols represent experimental data points, and solid lines show the corresponding best fits using the Hill equation.

Table 1 Measured surface tensions of polymer samples at different pressures at 200 °C
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To examine the generality of the observed behavior, 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 a molecular weight of 10,000 g/mol 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 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.

Fig. 3: Master curve of pressure dependence for polymer melts.
Fig. 3: Master curve of pressure dependence for polymer melts.The alternative text for this image may have been generated using AI.
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Scaled surface tension \((\frac{\gamma -{\gamma }_{0}}{{\gamma }_{\infty }-{\gamma }_{0}})\) versus \(\left(P/{P}_{0}\right)\) for PEG (blue filled circle), low molecular weight PEG (PEGLmw) (orange filled circle), PS (green filled circle), PI (red filled circle), PP (purple filled circle), PDMS (red filled circle), and low molecular weight PDMS (PDMSLmw) (light blue filled circle). Symbols denote scaled experimental data, and the solid curve shows the model using the Hill equation, demonstrating that all the data points fall on a master curve (solid blue line).

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 \({\gamma }_{0}\), and the contribution from interaction with the air molecules, \(\Delta \gamma\). The reduction of the surface tension in the high vacuum (low pressure) region is due to air-polymer interaction, which 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-neighbor interactions between the different species. For a surface in the xy plane, the surface area is \(S=N{a}^{2}\), where \(N\) is the number of lattice sites on the xy plane, and \(a\) is the lattice constant. The total interaction energy between the polymeric species and the lattice gas is given by

$$E={N}_{{PA}}{\varepsilon }_{{PA}}+{N}_{{PB}}{\varepsilon }_{{PB}},$$
(1)

where \({N}_{{PA}}\) and \({N}_{{PB}}\) are the number of nearest-neighbor polymer-air and polymer-vacuum bonds, respectively, and \({\varepsilon }_{{PA}}\) and \({\varepsilon }_{{PB}}\) are the corresponding interaction energies. We use a mean-field approximation to estimate the number of nearest-neighbor bonds. Specifically, we assume that the air concentration is given by \(\phi\), then the average number of nearest-neighbor bonds is given by \({N}_{{PA}}=\phi N\) and \({N}_{{PB}}=\left(1-\phi \right)N\). Therefore, the total surface energy becomes,

$$E=N\phi {\varepsilon }_{{PA}}+N\left(1-\phi \right){\varepsilon }_{{PB}}=N\left[{\varepsilon }_{{PB}}+\phi \left({\varepsilon }_{{PA}}-{\varepsilon }_{{PB}}\right)\right]$$
(2)

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,

$$\gamma=\frac{E}{S}=\frac{{\varepsilon }_{{PB}}+\phi \left({\varepsilon }_{{PA}}-{\varepsilon }_{{PB}}\right)}{{a}^{2}}$$
(3)

We now introduce the polymer-vacuum surface tension, \({\gamma }_{0}=\frac{{\varepsilon }_{{PB}}}{{a}^{2}}\), and the differential interaction energy \(\Delta \varepsilon={\varepsilon }_{{PA}}-{\varepsilon }_{{PB}}\), the surface tension is now given by,

$$\gamma={\gamma }_{0}+\frac{\Delta \varepsilon }{{l}^{2}}\,\phi$$
(4)

where \(l\) is a length scale characterizing monomer-monomer distance, \(\Delta \varepsilon\) quantifies the difference of air-polymer and vacuum-polymer interaction, and \(\phi\) 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 could be approximated by the ideal gas law, \(\phi=\frac{N{v}_{0}}{V}=\frac{{v}_{0}P}{{k}_{B}T}\), where \({v}_{0}\) is the volume of one gas molecule. Using this expression obtained,

$$\gamma={\gamma }_{0}+\frac{{v}_{0}\Delta \varepsilon }{{l}^{2}}\,\frac{P}{{k}_{B}T}$$
(5)

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 pressure2,12,14,15,16,17. The drop in surface tension in high-pressure conditions 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 behavior 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 Hill’s function31,32,

$$\phi=\frac{{P}^{H}}{{P}_{0}^{H}+{P}^{H}}$$
(6)

Here, \(H\) is the Hill coefficient, and \({P}_{0}\) represents a threshold pressure at which \(\phi=1/2\). The familiar Langmuir adsorption isothermal \((\phi=P/({P}_{0}+P))\) is recovered when H = 1. The Hill equation is used widely in biochemistry to describe ligand binding, which is similar to the adsorption process. The Hill coefficient is commonly found to be larger than \(1\) in biological systems, indicating positive cooperative 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,

$$\gamma={\gamma }_{0}+\left({\gamma }_{\infty }-{\gamma }_{0}\right)\frac{{\left(P/{P}_{0}\right)}^{H}}{1+{\left(P/{P}_{0}\right)}^{H}}$$
(7)

Here, \({\gamma }_{0}\), \({\gamma }_{\infty }\) and \({P}_{0}\) are fitting parameters for a fixed value of the Hill coefficient H. \({\gamma }_{0}\) is the surface tension in vacuum (P = 0), and \({\gamma }_{\infty }\) 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 a good fit to all the data. After some exploration, we decided to use \(H=0.2\), which gives a good fit 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 binding33. It is interesting to note that for finite value of \(P/{P}_{0}\) and small \(H\), the Hill function can be approximated as \(\phi \approx \frac{1}{2}(1+\frac{H}{2}{{\mathrm{ln}}}P/{P}_{0})\,\), therefore the observed surface tension data could be described by a linear function of \({{\mathrm{ln}}}P\). It is noted that the parameter \({P}_{0}\) in our model as shown in Eq. 7 can be temperature-dependent. When the Hill function is used, the temperature dependence of the surface tension is embedded in the parameter \({P}_{0}\) in our model. The temperature-dependence shown in Fig. 1 could be explained by using the approximate expression \(\phi \approx \frac{1}{2}(1+\frac{H}{2}{{\mathrm{ln}}}P/{P}_{0})\,\).

Table 2 Fitting parameters for the data points
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As compared to other polymer systems, the PI sample shows a different behavior in that the fitting parameter \({P}_{0}\) is much larger than that for all other samples. We speculate that this might be 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,

$$\frac{\gamma -{\gamma }_{0}}{{\gamma }_{\infty }-{\gamma }_{0}}=\frac{{\left(P/{P}_{0}\right)}^{H}}{1+{\left(P/{P}_{0}\right)}^{H}}$$
(8)

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 behavior of the surface tension as a function of air pressure in the low-pressure region should be universal, independent of the polymer samples. It is noted that air is a homogeneous mixture of N2, O2 and Ar. In our simple model, the contribution of different interactions of those gas molecules with the polymer is represented by an averaged interaction parameter. Certainly, the model could be improved by performing control experiments to differentiate the interaction between individual molecules and the 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 Jacobs et al. observed that the pressure effect on polymer thin film rupture and showed 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 in polymer surface tension when the gas pressure is decreased to below 1 mbar, independent of 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 the Hill function does provide an adequate description of the experiments.

Discussion

In conclusion, the surface tension of several polymeric melts has been measured by the pre-coated capillary technique, where the temperature and air pressure were controlled by using a homemade oven. At ambient conditions, the measured surface tension data exhibit a slight linear decrease when the temperature is increased, in agreement with the literature, validating our experimental protocol. Under high vacuum, it is discovered that the surface tension of polymers exhibits a drastic drop when the air pressure is decreased to below \({10}^{3}\,{{{\rm{N}}}}/{{{{\rm{m}}}}}^{2}\), which is opposite to the pressure effect at high-pressure region, where the surface tension exhibits a slight decrease when the pressure is increased. In the air pressure range of \({10}^{-4}\) to \({10}^{5}\,{{{\rm{N}}}}/{{{{\rm{m}}}}}^{2}\), the measured surface tension follows the Hill equation, \(\gamma={\gamma }_{0}+\left({\gamma }_{\infty }-{\gamma }_{0}\right)\frac{{\left(P/{P}_{0}\right)}^{H}}{1+{\left(P/{P}_{0}\right)}^{H}}\) with \(H=0.2\). A key limitation of our results is the limited number of data points at high vacuum conditions, which is mainly attributed to the difficulties in maintaining precise control over intermediate vacuum pressures and achieving ultra-high vacuum levels. The dataset can be expanded to include a broader range of polymer systems and enhance the precision of vacuum control, representing important directions for future investigation. Also, this discovery of the dramatic variation of surface tension in regular polymeric melts gives rise to new ideas for the fabrication of BCP thin films in a variety of applications such as nanopatterning. Furthermore, the observed pressure effect should not be specific to polymers and should be universally applicable to any physicochemical systems.

Methods

Materials

The polyethylene glycol (Mw = 20,000 g/mol, catalog 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 our laboratory. Polypropylene (Mw = 12,000 g/mol, product no. 428116) was a commercial product obtained from Sigma-Aldrich.

The polymer samples, whether commercially obtained or prepared in our 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 pumps 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 (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 Fig. 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.