Temperature dependence of crystal melt coexistence for supported polyethylene filaments

Domain: physics
Published: 2025-12-13
Editorial outcome: minor_revision

Abstract
An interface or surface may be considered as a planar perturbation reflected by changes in molecular properties in the direction perpendicular to the interface or surface. As a consequence, predicted by theory and shown by experiments, crystals are often covered by a thin liquid layer of their own melt. Such crystal–melt coexistence can be related to phenomena of surface premelting, secondary nucleation and melting point depression, particularly important for small systems. Here, we employed intermittent-contact mode atomic force microscopy imaging on nanoscopic semi-cylindrical filaments of polyethylene on a substrate to observe that these filaments contained a crystalline core bounded by molten regions of rather uniform width, W_soft = (9±2) nm at room temperature, which increased reversibly with temperature T. Filaments smaller than ca. 2*W_soft(T) were completely molten. The values of W_soft(T) compared favorably with theoretically predicted characteristic length scales in the context of nucleation, surface premelting and the melting point depression of finite size crystals. Altogether, we propose that these three phenomena are related and dominated by the intermolecular forces acting at crystal surfaces.

Introduction
Intermolecular interactions at surfaces, often characterized by the corresponding surface tensions, determine if a solid may be completely or partially wetted by its own melt1. In the complete wetting case, a surface starts to melt and forms a thin liquid layer at temperatures T well below the equilibrium bulk melting temperature T_m^1, a process typically called surface premelting (SM). In the partial wetting case, the solid remains fully crystalline up to and even beyond T_m^1 (superheating), referred to as surface nonmelting (NM)2. Interestingly, as shown for lead3,4 or gold2,5, different crystallographic faces of a single substance may show either SM3,5 or NM2,4. For SM, the thickness l of the wetting layer, characterized by the existence of stable interfaces between a crystalline core and a liquid phase at its surface, depends on T and approaches infinity at T_m^1 3,5,6. Thus, for a system of finite size, the crystalline core disappears once the liquid phase spans the whole system. This limiting size of the system, where no crystalline phase can exist, is related to the size of the corresponding critical nucleus7,8. Accordingly, neglecting the influence of geometry and curvature, the minimum size of a nanocrystal and the thickness of the layer of its own melt coating the crystal show similar functional dependence on T, both length scales grow to large values upon approaching T_m^1. Related to such crystal-melt coexistence are phenomena of melting point depression7,8, (secondary) nucleation9 and surface premelting3,5,6,10. In the past, the size- and temperature-dependence of these phenomena have been well studied, showing that the physics behind these three phenomena originates from the broken symmetry of intermolecular interactions at interfaces, typically discussed by the balance of an energy per surface (the surface term), and an energy per volume (the volume term). The ratio of surface term / volume term introduces a length scale l which reflects the key physical concept underlying all three phenomena. Several theoretical approaches11,12 were used to describe the temperature-dependence of these phenomena. Experimental investigations5,6,10,13 and a simultaneous comparison of the resulting observations in terms of nucleation, wetting and melting are rare, in particular for nanoscopic systems.

As shown by computer simulations14–16 for finite systems in a canonical ensemble (N, V, T), i.e., for a fixed number N of molecules and a finite volume V at a temperature T, it is possible to observe stable interfaces between a crystalline and a liquid phase that are in thermodynamic equilibrium. For example, in the absence of an external field or interactions with a substrate, a crystal can be surrounded by a fluid layer15–20. As an equilibrium state, such coexistence is stable in time and is re-established even after small perturbations. As further deduced from computer simulations16, this coexistence does not depend on the geometry of the system and the shape of the crystal. A connection to the size of a critical nucleus in large (infinite) systems has been drawn16. While the solid cluster (crystal) found in (N, V, T) simulations of finite systems is in equilibrium with the surrounding liquid, the same cluster in a grand-canonical ensemble (μ, V, T), where the chemical potential μ, V, and T are constant while the particle number N can vary, is unstable and has been identified as the critical nucleus.

In the study presented here, we experimentally explored the coexistence of a nanoscopic polymer crystal with a liquid (molten) layer characterized by a thickness, which we compared with predictions of the Lauritzen–Hoffman theory of secondary nucleation at the crystal growth front, represented by the complementary length scale of the critical nucleus. Our findings propose that, for nanoscopic systems, the temperature dependence of the characteristic length scales of various phenomena, i.e., the size of the critical nucleus, the thickness of the liquid layer in surface premelting, and the size-dependence of the melting-point depression of finite-size crystals, are related. To generate separated polymer crystals of small size, we required a system with a highly limited number of polymer chains. We employed spin-coated ultra-thin films of polyethylene (PE), a well-investigated polymer21,22. PE is a non volatile polymer, which ensured that the total number of molecules in our sample was a conserved quantity. Using solutions of low polymer concentration, the areal polymer density (represented by a mean film thickness) of these samples was extremely low. Previous experiments23 have shown that polymer chains in ultra-thin layers, often prepared by spin coating21,23, crystallize very slowly. Sometimes, the crystallization process was so slow that no crystals were detectable within the provided experimental time window of hours or even days23. An extremely low nucleation rate and slow surface diffusion caused by attractive interactions with the substrate were invoked as explanations for these observations24. In an undercooled melt, crystals can only grow if a heterogeneous or a homogeneous nucleus is established first24. However, due to the limited number of polymer chains available in such ultra-thin films, the probability for generating such nuclei is extremely low24,25. As a result, typically only very few homogeneous (primary) nucleation events, or none at all, occur within the whole sample within the provided experimental waiting time23,24.

Fortunately, the homogeneous nucleation step may be circumvented by introducing filaments consisting of stretched and uniquely oriented polymer chains26,27. As previously demonstrated28 through systematic spin coating experiments with dilute polyethylene (PE) solutions, a shear-induced change in polymer chain conformation can occur in the corresponding flow field caused by spin coating, i.e., polymer chains undergo a transition from randomly coiled to stretched chains. As a consequence of the resulting attractive interactions between stretched polymer chains, bundles of PE chains are formed28–31. Coalescence and fusion of bundles result in the formation of ultra-long filaments28. Due to the massive loss of conformational entropy and the high degree of order of the polymer chains within the filaments (often called “shish”)32, polymer chains may crystallize easily within the filament33. However, during sample preparation, the flow conditions induced by the spinning substrate varied in space and time28. Thus, not all polymer chains can undergo such a coil-stretch transition. Consequently, in the resulting as-prepared sample, filaments of stretched and crystalline chains were surrounded initially by non-stretched molten polymer chains.

Filaments can act as sites for secondary nucleation27,32, circumventing the problem of the low homogeneous nucleation rate of ultra-thin polymer films and allowing the growth of polymer crystals in such films, often resulting in the formation of so-called “shish-kebab”-patterns27,32,33. However, to initiate such growth, first a secondary nucleus need to form on the filament, which requires that the number of available polymer chains is above a threshold value. Furthermore, along the same line of reasoning, we anticipate that crystal growth will stop once the number of remaining molten polymer chains is below this threshold value. For crystals growing on filaments, the polymer chain axis in such crystals is easily identifiable, being identical with the long axis of the filaments. Thus, filaments introduce a unique orientation and define the growth direction for the formation of lamellar polymer crystals, often called “kebabs”32–34, which will always grow perpendicular to the filament.

A first set of experiments on films of PE was performed at room temperature. This temperature is well below the melting temperature but above the glass transition temperature of PE, allowing for sufficient mobility of non-crystallized polymer chains but also for crystal growth. Furthermore, the adsorption energy of PE on the chosen atomically-smooth and non-epitaxial substrate, i.e., mica, is sufficiently weak not to impede surface diffusion of PE. Thus, all polymer chains had a chance to crystallize. As predicted by theoretical considerations on solid-liquid coexistence14,15,35, the crystalline core was covered by a layer of molten polymer which was separated from the core by a well-defined interface. To investigate this crystal–melt coexistence, we analyzed the mechanical properties of filaments and their surrounding either crystalline or molten polymer via the phase signal of intermittent-contact mode atomic force microscopy (AFM)36–45. Furthermore, we determined the temperature-dependence of the thickness of the liquid layer on a crystal surface for a large range of T below T_m^1.

Results
As deduced from their long lifetime of many months, the investigated PE filaments were partially solid. If these filaments had contained molten polymer chains only, they would have decayed into droplets via the Rayleigh-Plateau instability46. Naively, one might expect that filaments have uniform mechanical properties of a homogeneous crystalline solid. However, the phase signal derived from the intermittent-contact mode AFM measurements clearly revealed the coexistence of two regions across the filaments, characterized by distinctly different mechanical properties. Remarkably, this observation was independent of the height and diameter of the filaments and similar for both types of studied filaments, as revealed by comparing Fig. 1a and b with Fig. 1c and d. Measurements of the same samples repeated after up to several months of storage at room temperature confirmed that the contrast in the AFM signal for the two regions within the filaments did not change during the time of storage. Thus, we conclude that the observed coexistence of the two regions was stable in time and probably represents an equilibrium pattern.

Clearly, the “hard” region (with a high phase value, thus a high elastic modulus (E)) of a filament was bounded symmetrically on both sides by “soft” regions (with a low phase value, thus a low E) of similar width (see Fig. 1b and d), labeled as W_soft. In section 2 of the Supplementary Information we provide more evidence about the coexistence of “soft” and “hard” regions and their identification. From the cross-sections shown in Fig. 1e, we derived the value of W_soft as the full width at half maximum of the peak in the phase signal (see Fig. 1e).

We performed systematic AFM measurements on a single PE filament using intermittent-contact mode with a decreasing setpoint value, thus a decreasing setpoint ratio (s)44 (the definition and details of s are described in the “Methods” section.) Depending on the force (f_T) applied via the AFM tip onto the sample, the AFM cantilever tip penetrates the material on the sample surface38,43,45. For small values of f_T, the depth of penetration (d_p) at position (x, y) into the sample also depends on E of the sample36. Thus, d_p is recorded as d_p(x, y, f_T, E). In Fig. 1a, the directions of x and y are defined. For a given AFM cantilever tip with a fixed shape, for small values of f_T and for a sample with time-independent mechanical properties, the measured deformation of the sample increases linearly with increasing f_T 40. Deformation is often visualized as a change in topographical features of the sample, expressed by changes in the measured height data. As a result, the measured height H^ (x, y, f_T, E) deviates from the true surface topography H(x, y) of the sample by the depth of penetration d_p(x, y, f_T, E). By decreasing s, i.e., increasing f_T, we get: H^ (x, y, f_T, E) = H(x, y) - d_p(x, y, f_T, E). Therefore, with increasing f_T, we observed a decrease in H^ (x, y, f_T, E), which was larger in the soft regions than in the hard regions. (Fig. 2a, b). For the conditions applied here, in particular for the small values of d_p, the area of contact between tip and sample is expected to vary only slightly. Therefore, in parallel to the increase in f_T, we also have an increase of the applied pressure.

Besides, once the AFM cantilever tip pierced through the whole thickness of the soft region, i.e., yielding eventually H^ (x, y, f_T, E) ≈ 0 nm in this soft region, the correspondingly observed phase value was progressively dominated by the properties of the underlying mica (see Fig. 2c).

The filaments investigated here consisted only of a single polymer, i.e., polyethylene (PE). All measurements shown in Figs. 1, 2 were performed at the same temperature, i.e., at room temperature (ca. 20 °C), which is between the glass transition temperature and the melting temperature of PE. Thus, the large difference in the values of the intermittent-contact mode phase signal demonstrates that, even at room temperature, within each PE filament, a solid crystalline core coexisted with meniscus edges of molten polymer, i.e., soft zones at the lateral boundary of the filament.

As shown in Figs. 1, 2, and sketched in Fig. 3, filaments were covered by a region of soft molten polymer. Here, we consider filaments as long semi-cylinders on a planar substrate. Due to the semi-cylindrical geometry of the nanofilament and its interaction with the wettable mica substrate, we have to deal with different values of the interfacial energies and surface tensions across the filament. One possibility is that not all parts of the filament are covered with a liquid layer, and the conditions for surface melting (or non-melting) are not fulfilled for all parts of the nanofilament. The other possibility is that an ultrathin liquid layer exists at the top of the filament. In the latter case, it is so thin that its AFM phase signal is dominated by the mechanical response of the underlying crystalline material, possibly with a minor (insignificant) contribution from this ultrathin surface liquid layer. In contrast, the meniscus region exhibits a much greater effective thickness. There, the AFM phase response is clearly dominated by the properties of the molten polyethylene—at least until the tip fully penetrates this region. Thus, the observed soft regions in the AFM phase images correspond to the molten polymer chains in the concave meniscus region at the edges. In section 2 of the Supplementary Information, we provide a set of force spectroscopy measurements taken at successive positions along the cross-section of a filament, supporting the sketch shown in Fig. 3a.

We approximated the cross-section of the soft region by a concave shape (see Fig. 3a). The concave outline of this wedge-shaped meniscus region is favoring capillary condensation, as reflected through the Kelvin equation47–49. Thus, as a consequence of the acting capillary forces, polymer chains accumulated in the wedge, forming a liquid-like zone near the (filament-substrate-air) three-phase-contact-line. The number of polymer chains which potentially could diffuse to the wedge-shaped meniscus region depended on the number density of polymer chains adsorbed on the substrate surrounding the filament. The surface of the meniscus region is characterized by a constant radius of curvature R_?. We note that the curvature in the direction along the filament is approximately zero (i.e., R_|| ≈ ∞). Accordingly, as a consequence of the surrounding atmospheric pressure p_atm, i.e., the Laplace pressure was negative, Δp_Laplace = p_wedge - p_atm < 0. A negative Laplace pressure, which was constant (R_? = constant) within the entire soft region along the whole filament, assured that thickness fluctuations were dampened. Thus, within some small fluctuations, the height and the width of the soft region remained constant along the whole filament.

The fact that we have observed on the same spin-coated sample two types of straight filaments, some with a uniform width and others decorated with “beads-on-a-string” patterns, suggests the existence of a threshold number of polymer chains that can be included in the meniscus region. Beyond this threshold value, secondary nucleation of crystalline structures becomes possible. Such nucleation events lead to the formation of so-called “shish-kebab” crystalline patterns, characterized by edge-on lamellar crystals growing perpendicular to the long axis of the filament. As demonstrated previously for this type of sample preparation based on spin coating, filaments containing stretched polymer chains are co-deposited together with non-stretched polymer chains. The latter forms an adsorbed ultrathin layer of liquid-like polymer in the surroundings of the filaments. We assume that filaments acted as attractors for the surrounding molten non-stretched chains. Eventually, these chains form a liquid meniscus-shaped zone around the filaments. We further assume that, for a width larger than W_soft, these soft regions become unstable. Possibly caused by a kind of Plateau–Rayleigh-type instability, the liquid meniscus region is transformed into many bead-like structures on the filaments, thereby generating a “beads-on-a-string” pattern. Given that the size of the beads is larger than the critical size of a crystalline nucleus, the beads partially crystallized. Deduced from the corresponding intermittent-contact mode phase signal, these beads also consisted of a hard core bounded by a soft region. In section 3 of the Supplementary Information we provide AFM images for filaments with beads-on-a-string patterns.

Given that the surface of the core of a filament (often called shish) consisted of locally ordered, crystalline chains, epitaxy may allow for the induction of secondary nucleation followed by the growth of edge-on lamellar crystals (often called kebabs). However, formation of kebabs is only possible if a sufficient number of polymer chains is available at its growth site. A secondary nucleus requires a certain number of crystalline stems of a certain length which can be derived from a single or multiple polymer chains50,51. We note that an analogy but also some differences exist to observations of substrate-induced prefreezing of polymer chains, for example, on graphite19,20. In both cases, we observe stable coexistence of crystalline and molten polymer chains in the surrounding and at the growth front is below this minimum value, we anticipate that crystal growth will stop. Accordingly, if the residual number of molten polymer chains at the growth front per appropriate length along the filament adds up a total volume less than V_cr, these polymer chains cannot crystallize and will stay in the molten state. Moreover, since the secondary nuclei adopt the same orientation as the original crystal—that is, the nanofilament in which the polymer chains are aligned along its axis—the cylindrical nucleus is aligned lengthwise along the nanofilament, while its radius is oriented perpendicularly; this radius is directly correlated with the value of W_soft, i.e., W_soft ≈ r*. This implies that the crystal-melt coexistence observed at the growth front of the polymer crystal differs from the well-studied coexistence at the fold surface. Consequently, we anticipate that filaments with a crystalline core (blue area in Fig. 3a) are surrounded with an narrow zone of molten polymer at the growth front (red area in Fig. 3a). In order to support our interpretation, we have superposed the sketch of Fig. 3a with the measured cross-sections of the AFM height and phase data shown in Fig. 1e (see Fig. 3b). There, the phase signal switches from negative values at the periphery of the cross-section to a positive value in the center of the filament, representing the soft zones of molten polymer at the periphery and the hard (crystalline) core of the filament, respectively.

Invoking the concept of secondary nucleation for the soft regions at the periphery of the filament and employing CNT, we arrive at two predictions for the equilibrium width W_soft of the soft zone of molten polymer. First, at a given temperature, employing thermodynamic arguments inherent to CNT, we assume that a cylindrical secondary nucleus can only form in the soft zone of molten polymer at the periphery of a hard core of a filament when the width W of this zone is larger than the size of the critical secondary nucleus, characterized by r*, i.e., we assume W_soft ≈ r*. For fixed experimental conditions, we expect that the size r* of the critical secondary nucleus is constant and should not depend on crystal size or morphology, i.e., filaments having either a uniform width or being decorated with beads-on-a-string patterns should yield the same value of W_soft. In fact, all values of W_soft shown in Fig. 4 were similar and narrowly distributed around (9±2) nm. Accordingly, within error bars, the observation of a constant value of W_soft is consistent with the assumption of W_soft ≈ r*.

For long polymer chains, r* is independent of polymer molecular weight52. Thus, the dispersity of the polymer is not expected to influence either r* or W_soft. Furthermore, for samples prepared on the same type of substrate (here: mica), i.e., for identical wetting conditions related to the surface tension of the substrate and the polymer, we may assume that the geometry of the soft meniscus region containing molten polymer is the same for all filaments, independent of their size or crystal morphology. In fact, for the two PE samples with largely different dispersity in chain length, our experimental results yielded similar values of W_soft.

The second prediction based on the concept of secondary nucleation concerns the temperature dependence of W_soft(T), which is expected to be proportional to r*(T). According to CNT, the temperature dependence of r* can be derived from the following equation9:

r* = 2*σ_s / ΔG_V (1)

where ΔG_V is the free energy change, per unit volume, involved in the formation of a crystal nucleus, σ_s is interfacial or surface energy of the nucleus.

ΔG_V can be approximated by ref. 9:

ΔG_V ≈ ρ_m * ΔH_f * (T_m^1 - T) / T_m^1 (2)

where T is the experimental temperature of the sample (filament), ΔT = T_m^1 - T, ρ_m is the mass density of polymer, ΔH_f is the heat of fusion per unit mass of the polymer at T_m^1 = 141 °C.

Following refs. 53,54, we used the following values for PE: σ_s ranges from 12 mJ/m^2 to 15 mJ/m^2; ΔH_f ≈ 4.1 ± 0.2 kJ/mol of CH_2, and ρ_m is ranging from ca. 780 kg/m^3 to ca. 970 kg/m^3. Based on Eq. (2), we obtain r*(T) or analogously r*(ΔT):

r*(ΔT) ≈ P * (T_m^1 / ΔT) (3)

where P = (2 * σ_s) / (ρ_m * ΔH_f) = (0.10 ± 0.02) nm. Furthermore, as a consequence of the thermodynamic arguments underlying CNT, changes of r* induced by varying T are expected to be reversible. Based on the relation W_soft ≈ r*, we anticipate that W_soft(T) should become larger at higher temperatures and should become smaller again when T was reduced back to low temperatures, e.g., to room temperature.

To support our hypothesis of a relation between secondary nucleation and the measured values of W_soft(T), we performed a series of AFM measurements at different temperatures T. We employed the following sequence of temperatures: Room temperature (ca. 20 °C) → 100 °C → 80 °C → 60 °C → back to room temperature. As can be seen by comparing Fig. 5a, b, the width of the soft zone was clearly larger at the higher T. From a series of experiments, the statistical analysis of W_soft showed a decrease as a function of increasing degree of supercooling ΔT. On a multitude of samples, measurements were performed on various filaments, with either a uniform width or they were decorated with beads-on-a-string patterns. Most importantly, this increase of W_soft with T was fully reversible as demonstrated by the values of W_soft obtained at room temperature, measured before and after heating the sample to 100 °C, represented by the black and green data points in Fig. 5c. In section 5 of the Supplementary Information we provide additional AFM images supporting the reversibility of the changes of W_soft(T).

When comparing r*(ΔT) with the experimentally measured values of W_soft(ΔT), because we have to account for the specific geometry of the wedge-shaped soft zone of molten polymer and the influence of the mica substrate, W_soft(ΔT) ≠ r*(ΔT), but only proportional to r*(ΔT), i.e., W_soft(ΔT) ≈ r*(ΔT). Thus, a constant prefactor C_CNT is required to relate W_soft(ΔT) with r*(ΔT), we obtain W_soft(ΔT) = C_CNT * r*(ΔT). Using C_CNT = 16 and employing Eq. (3) for r*(ΔT) yields the red line shown in Fig. 5c. A good match with the experimentally determined data points is observed. Thus, we can conclude that all filaments, including filaments decorated with bead-like structures, were bounded by coexisting soft zones of molten polymer of a size just below a size proportional to r*(ΔT) of the critical secondary nucleus.

Besides the limiting size derived from the minimum volume of molten polymer required for the formation of a stable (critical) secondary nucleus at the crystal growth front, the coexistence of a liquid layer on a crystal is also theoretically predicted6,11,13 and experimentally verified6,11,13 for planar surfaces on bulk samples in the context of surface premelting. Assuming a specific distance-dependence of intermolecular interactions, various functional forms of the temperature variation of the characteristic length scale, represented as l, of a liquid layer on the surface of a crystal have been predicted (see Table 1). However, these approaches did not account for the influence of geometry, curvature, and the influence of the mica substrate, making a direct comparison with our results problematic. Nonetheless, as seen by the blue lines in Fig. 5c, within the uncertainty of our data points of W_soft(ΔT), a description along the lines of surface melting is possible.

From Fig. 5, we can deduce that the increase of W_soft with T indicates the progressive melting of PE crystals at their surface, suggesting that melting of PE crystals is a continuous process55. Interestingly, for a given finite size of a filament, characterized by its width w, such an increase of W_soft(T) allows to define a size-dependent melting temperature of a filament T_m(w) ≤ T_m(w=∞) = T_m^1, a phenomenon often referred to as size-dependent melting point depression of nanostructures7,8. In other words, when 2 * W_soft(T) ≈ w, then the whole filament is molten. Accordingly, thinner (narrower) filaments melt at lower temperatures than wider ones. The corresponding temperature dependence of T_m(w) ≈ ΔT^-1 is described by the Gibbs-Thomson (GT) equation7,56 (see Table 1). Some typical AFM phase images in support of the melting point depression in nanoscopic filaments are shown as insets in Fig. 5c. The GT-equation is represented by the black line in Fig. 5c, which can be interpreted as a phase boundary between the region where nanocrystals do not exist – they are completely molten [for w ≲ 2 * W_soft(T)] – and the region where (nanoscopic) crystalline core coexists with a molten surface layer [for w ≳ 2 * W_soft(T)]. In section 5 and 7 of the Supplementary Information, we provide additional AFM images in support of the melting point depression in nanoscopic filaments. As shown previously57, even after the complete melting of the nanoscopic filaments, the stretched polymer chains inside the filament kept some memory which decayed only slowly.

As described and discussed above, we can relate our experimental observations of crystal–melt coexistence and the corresponding temperature dependence of W_soft with three different phenomena and the corresponding characteristic length scales: the size of the critical secondary nucleus, the thickness of the molten wetting layer and the size-dependent melting point depression of nanostructures. Although these phenomena were often treated independently, our results highlight their fundamental relationship. All three phenomena originate from the broken symmetry of intermolecular interactions at surfaces and the corresponding competition between interfacial energy and internal energy.

Methods
Materials and sample preparation
For the present study, we employed long polymer filaments of PE obtained by spin coating dilute solutions of PE in para-xylene (PX, Sigma-Aldrich)28. Most filaments were prepared from a 0.075 wt% dilute solution of PE in PX, which was heated to ca. 130 °C in a thermostat. This solution was spin coated onto a ca. 80 °C hot and freshly cleaved mica substrate (Plano GmbH, Wetzlar/Germany) rotating at 8000 rpm (using an acceleration period of 1 second for the spin-up stage). Two different PE samples were examined: PE-M, consisting of PE with a weight-average molecular weight M_w = 1800 kg/mol and a dispersity Ɖ = 3.1 (Institut für Makromolekulare Chemie, Freiburg/Germany), and PE-B, representing a blend of 50 wt% of PE-M and 50 wt% PE with M_w = 277.3 kg/mol and a Ɖ = 27.4 (Sinopec Research Institute of Petroleum Processing). Spin coating experiments were performed with a device from Laurell Technologies Corporation, Lansdale, PA, USA (Model WS-650MZ-23NPP).

Characterization techniques
Topography and mechanical properties of all samples were determined with the help of an atomic force microscope (AFM, NanoWizard II from JPK Instruments, Berlin/Germany). For all measurements, operated in the intermittent-contact mode in air, we used Tap190Al-G AFM cantilever tips from Budget Sensors, Bulgaria with a characteristic radius of curvature of around 10 nm, a characteristic spring constant of the cantilever of around 48 N/m, and a resonance frequency of the cantilever of around 165 kHz. The scan angle did not have any detectable influence on the resulting images. The values of the phase-signal derived in the AFM intermittent-contact mode allowed to detect differences in elastic properties of the measured sample36,39,42,43,45. For our AFM measurements, which were performed in the repulsive regime, we chose a parameter setting where a lower phase value referred to a lower elastic modulus36. For both, phase and height data, the value characterizing the surrounding mica substrate (the background) was used as a reference, which was set to a value of zero. Therefore, the “soft zone” is defined as the area with a negative phase value, high energy dissipation; the “hard core” of filaments is defined as the area with a positive phase value, i.e., low energy dissipation36,42,58.

For the in-situ high-temperature AFM measurements, we used a High-Temperature Heating System (HTHSTM) from JPK Instruments, which we calibrated with a K-type thermocouple before each measurement. To allow for thermal equilibration after each step-wise change in temperature, the sample was kept at the set temperature for more than 30 minutes before starting the measurement.

During a typical intermittent-contact mode AFM measurement (in air), the AFM cantilever oscillates at a frequency, which is in the order of 100 Hz lower than one of the resonance frequencies, with a “free” amplitude A_0, obtained at a position where the cantilever tip was at least 100 μm from the sample surface36,59. When the cantilever approaches the sample surface so closely that attractive van der Waals interactions between AFM cantilever tip and sample start to become significant, the oscillation amplitude may increase, but subsequently, at even closer distances to the surface, will decrease below A_0 due to repulsive interactions and some deformation of the sample induced by the force applied through the cantilever. The force applied to the sample surface is proportional to the spring constant and the deflection of the cantilever. An electronic feedback loop (in this paper, I-gain is 0.02, P-gain is between 30 Hz to 50 Hz) acts on the piezoelectric scanner of the AFM, providing precise control over the oscillation amplitude of the cantilever. During scanning of the AFM cantilever tip across the sample, the movement of the piezoelectric scanner in the vertical direction is adjusted in order to keep the oscillation amplitude of the cantilever at a pre-set value A, the so-called setpoint, that is slightly below the oscillation amplitude (A_0 = 73 - 80 nm)38. (A more detailed discussion about the estimation of A_0 can be found in Section 6 of the Supplementary Information) Therefore, in this constant amplitude mode, AFM images were acquired at a constant value of the setpoint ratios, with s = A/A_0.

For thin layers and topographical features with small values H(x, y) on a substrate of high elastic modulus, we have to avoid that the penetration depth d_p is larger than H(x, y) or the layer thickness. Thus, in the here presented experiments, we have chosen the lowest experimentally accessible forces, i.e., we have chosen values of the setpoint ratio close to s = 138.

Data availability
The findings of this study are fully documented by the data provided in the manuscript and the supplementary information. The plotted and statistical data generated in this study have been deposited in the supplementary Excel file entitled: Original data-crystal melt coexistence. All data are available from the corresponding author upon request. Source data are provided with this paper.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "Review attached.\n[Editorial Note: See end of file]", "reviewer_2": "Please see the attached Word file with my 3.5-page report, which includes symbols and subscripts that are not transferred\ncorrectly when I paste the report into this window.\n[Editorial Note: See end of file]", "reviewer_3": "This paper presents an interesting study on the coexistence of a crystal- melt interface on supported polyethylene filaments.\nI found that the paper is not easy to read, and at times it is confusing. I urge the authors to revise it and make it more\nunderstandable.\nThe first phrase of the abstract is quite cryptic and not well written: \"\nThe broken symmetry of molecular interactions at interfaces is causing that crystals are often\ncovered by a thin liquid layer of their own melt.\" No explanation is given to what they refere as broken symmetry and they\ncannot use the verb in this tense \"is causing\", as this denotes that is happening at this instant...Consider revising and\nrewriting the abstract to indicate clearly what has been done and what is new and a significant advanced with respect t\nexisting literature.\nThe write: For all studied PE filaments and independent of their size, the coexistence of a hard (solid) core bounded by a soft\n(liquid-like) layer near the filament-substrate contact line was confirmed.\nAre they sure this holds for any PE filament size? Are we to expect a molten surface in a high molecular weight microscopic-\nsized PE fiber or bundles of fibers to have the same molten surface.\nIt would seem that the phenomena observed in most likely present in nanometer fibers.\nIn many instances, the authors use the word \"polymer\" to refer to polymer chains, for instance, just one example:\n\"However, in finite systems like ultra-thin nanometer-sized filaments the number of available molten polymers is easily\nexhausted. Thus, once the number of molten polymers in the surrounding and at the growth front is below this minimum\nvalue, we anticipate that crystal growth will stop.\"\nThere is no way they can use the expression \"number of available molten polymers is easily exhausted.\" I imagine they are\nreferring to the available molten chains; otherwise, I do not understand their meaning. This happens several times in the\npaper.\nThe experiments performed and the analyses presented are very good, but unfortunately, the paper is in many parts not\nclear enough, and the writing should be improved.\nThey have employed only PE filaments, but in principle, they could have used other filaments where they can easily vary the\ndegree of crystallinity and check its influence (i.e., PET).\nThe intermolecular interactions in this case are weak Van Der Waal forces (as only PE was employed). What would\nhappened if a polymer like PEO is used or PCL, in which the intermolecular forces are stronger?\nIs there any molecular weight dependence of the effects observed?"}, "author_response": "Detailed response on comments (written in blue) from Reviewer #1\nThe authors, based on intermittent AFM imaging, reported the coexistence of crystalline phase\nand molten phase of PE. The sample preparation and AFM imaging were carefully performed\nand therefore I can find some value in this manuscript, while I found some critical error in\ninterpreting their AFM results. Thus, I cannot recommend for the manuscript in the current\nform to be published in this journal.\nFirst and foremost, we extend our sincere gratitude to Reviewer #1 for the time and effort dedicated\nto reviewing our manuscript, as well as for providing highly valuable, insightful and constructive\nfeedback. We are profoundly encouraged by the positive assessment of our experiments. We fully\nacknowledge that the interpretation of our AFM data, particularly in the Supplementary\nInformation, was not sufficiently elaborated. Accordingly, to address all the points raised by\nReviewer #1, we have incorporated a more comprehensive discussion of these results in the revised\nmanuscript.\nJudging from the temperature-dependent change in 𝑊 , the authors caught some realistic\nsoft\nchange of those soft molten regions. However, judging from force-distance curves in Fig. SI-4,\nI found that the situation is not so simple.\nWe sincerely thank the reviewer for this insightful observation and for recognizing that the changes\ncaptured by the temperature-dependent variation in 𝑊 are realistic. We agree with the comment\nsoft\nthat the force–distance curves presented in Fig. SI-4 reveal additional complexity in the\nmechanical behavior of the soft molten regions. As this “extra” data was exclusively included in\nthe Supplementary Information, we did not provide a comprehensive or detailed discussion of it\nin the main text. In the following section, in response to the reviewer’s comment, we address in\nmore detail the complexity in the mechanical behavior by incorporating a thorough analysis and\nan extended discussion of the force–distance curves presented in Fig. SI-4.\nobtained on the mica substrate, while those are not the shape of curves on pure mica. After the\njump-in contact, there is a region with almost no change in force value (solid ellipse). This type\nof regions always observed when surface (or AFM probe itself) is contaminated. This\ncontamination can be the molten state PE. It also can be adsorbed water molecules on mica. Or,\nsomething else. Or, the intercalation of solvent molecules happened during spin-coating. The\nprobe surface can also be contaminated. The similar doubt was deduced from the withdrawing\ncurve. The long tail (dashed ellipse) will never appear for the contact between pure mica and\nclean AFM tip.\nWe sincerely thank the reviewer for the thorough and insightful comments regarding the force–\ndistance curves shown in Fig. SI-4. We agree that the shapes of these curves deviate from those\ntypically acquired on pure mica. We appreciate the reviewer’s careful analysis. In response, we\nprovide the following interpretation of the observed phenomena:\nAs shown in Fig. R1.1-1, the \"nearly-constant force region\" (indicated by a solid ellipse) and the\n\"extended tail region\" (indicated by a dashed ellipse) are observed on the deflection–distance\nmeasurement on a freshly cleaved mica surface with a new cantilever having a clean AFM tip (no\ncontamination). We attribute the apparent anomalies in the curves to the formation of a water\nmeniscus between the AFM tip and the sample. Under ambient conditions, the water meniscus is\ncaused by Kelvin condensation, which occurs universally. Thus, we observe water menisci also\n3\nfor the PE filaments, providing a consistent explanation for the nearly-constant force region (solid\nellipse) observed immediately after the jump-to-contact. Similarly, the extended tail visible during\nretraction (dashed ellipse) is a characteristic feature of contact with a water meniscus which\neventually ruptures.\nFigure R1.1-1: Deflection-distance traces taken with the same PPP-NCSTR cantilever tip. The\nmeasurement was first done on a freshly cleaved mica surface using a new AFM cantilever tip\n(top). Then, with the same cantilever tip, measurement was done on the region of the mica substrate\nsurrounding our UHMWPE filaments (bottom). During both measurements, the surrounding\nhumidity was 34% RH, and the temperature was 22 °C.\nFurthermore, a comparison between deflection–distance traces obtained on freshly cleaved mica\nand on the region of the mica substrate surrounding our UHMWPE filaments revealed a notably\nnarrower region of a nearly-constant force on the pure mica substrate. This difference suggests the\n4\npresence of an extremely thin, uniform layer of molten UHMWPE covering the mica surface in\nour samples. The uniformity of this molten UHMWPE layer is demonstrated by the homogeneous\nvalue of the AFM phase within the whole region of the mica substrate surrounding our UHMWPE\nfilaments, which was observed consistently throughout our study. The existence of such an\nultrathin layer of molten UHMWPE is expected due to the strongly different values of the surface\ntension of mica and polyethylene, lowering the surface tension of the region of the mica substrate\nsurrounding our UHMWPE filaments. Additionally, a pronounced jump in the deflection of the\ncantilever (marked by the red arrows in Fig. R1.1-1) is evident in both approach and retraction\ntraces on fresh mica, but appears only in the retraction direction of the curves measured on the\nregion of the mica substrate surrounding our UHMWPE filaments. This discrepancy occurs\nbecause, as the tip approached the surface, the ultrathin layer of molten UHMWPE initially\n“shields” the tip from strong attractive interactions with the actual (bare) mica surface. However,\nas this layer was penetrated during further extension, upon subsequent retraction the trace\nexhibited a sudden detachment event when the tip lost contact with this layer. Moreover, due to\nthe ultrathin layer of molten UHMWPE, this detachment \"jump\" during retraction is less sharp on\nthe UHMWPE-covered mica than on freshly cleaved mica.\nWe appreciate the reviewer’s concern regarding the potential influence of residual solvent\nmolecules on the sample surface. We would like to clarify that, although the presence of trace\nresidual solvent cannot be ruled out entirely, any such remaining amount is expected to be small\nand probably negligible. The spin-coating process employed para-xylene, a highly volatile solvent\nthat evaporates readily under ambient conditions. Furthermore, many of the samples have been\nstored unsealed in ambient atmosphere for over six months before performing measurements, with\nconstant air exchange ensured by continuous ventilation. Under these conditions, we are confident\nthat residual solvent effects, if any, do not affect the interpretation of our results.\nMoreover, the homogeneous and uniform signal observed in the AFM phase images of the whole\nregion of the mica substrate surrounding our UHMWPE filaments supports the robustness of our\ndata. Even if some level of cantilever tip contamination were present, its influence would manifest\nas a systematic offset in the phase values rather than as random or selective artifacts. Therefore,\nwhile we acknowledge the possibility of the cantilever tip getting contaminated by the UHMWPE\nmelt, such contributions would not alter the central conclusions drawn from our experimental\n5\nobservations. We appreciate the reviewer’s astute observations, which have helped us clarify these\nimportant mechanistic details.\nThe following information has been added at the end of section 2 of the Supplementary\nInformation.\nFig. SI-5: Deflection-distance traces taken with the same PPP-NCSTR cantilever tip. The\nmeasurement was first done on a freshly cleaved mica surface using a new AFM cantilever tip\n(top). Then, with the same cantilever tip, measurement was done on the region of the mica substrate\nsurrounding our UHMWPE filaments (bottom). During both measurements, the surrounding\nhumidity was 34% RH, and the temperature was 22 °C.\nIn force spectroscopy measurements conducted on the region of the mica substrate surrounding\nour UHMWPE filaments — specifically, lines 1, 2, and 10 in Fig. SI-4(a) — a distinctive feature\nwas observed: an area with nearly constant force value following the jump-in to contact (indicated\nby a solid ellipse) accompanied by a long tail in the retraction curve (indicated by a dashed ellipse).\nPossible sources for this liquid medium include a water meniscus formed via Kelvin condensation,\na thin layer of molten polyethylene (PE), or contamination of the cantilever tip by molten PE chains.\nTo identify the nature of this liquid medium, deflection–distance measurements were performed\nusing a new cantilever having a clean AFM tip (no contamination) on a freshly cleaved mica\nsurface. The same cantilever tip was then used to perform measurements on the mica region of the\nUHMWPE sample. As shown in Fig. SI-5, the \"nearly-constant force region\" (indicated by a solid\nellipse) and the \"extended tail region\" (indicated by a dashed ellipse) were observed on the\ndeflection–distance measurement on a freshly cleaved mica surface with a new cantilever having\na clean AFM tip (no contamination). We attribute the apparent anomalies in the curves to the\nformation of a water meniscus between the AFM tip and the sample. Under ambient conditions,\nthe water meniscus is caused by Kelvin condensation, which occurs universally. Thus, we\nobserved water menisci also for the PE filaments, providing a consistent explanation for the nearly-\nconstant force region (solid ellipse) observed immediately after the jump-to-contact. Similarly, the\nextended tail visible during retraction (dashed ellipse) is a characteristic feature of contact with a\nwater meniscus which eventually ruptures.\nFurthermore, a comparison between deflection–distance traces obtained on freshly cleaved mica\nand on the region of the mica substrate surrounding our UHMWPE filaments revealed a notably\nnarrower region of a nearly-constant force on the pure mica substrate. This difference suggests the\npresence of an extremely thin, uniform layer of molten UHMWPE covering the mica surface in\nour samples. The uniformity of this molten UHMWPE layer is demonstrated by the homogeneous\nvalue of the AFM phase within the whole region of the mica substrate surrounding our UHMWPE\nfilaments, which was observed consistently throughout our study. The existence of such an\nultrathin layer of molten UHMWPE is expected due to the strongly different values of the surface\ntension of mica and polyethylene, lowering the surface tension of the region of the mica substrate\nsurrounding our UHMWPE filaments. Additionally, a pronounced jump in the deflection of the\ncantilever (marked by the red arrows in Fig. SI-5) is evident in both approach and retraction traces\non fresh mica, but appears only in the retraction direction of the curves measured on the region of\nthe mica substrate surrounding our UHMWPE filaments. This discrepancy occurs because, as the\nAFM tip approached the surface, the ultrathin layer of molten UHMWPE initially “shields” the tip\n7\nfrom strong attractive interactions with the actual (bare) mica surface. However, as this layer was\npenetrated during further extension, upon subsequent retraction the trace exhibited a sudden\ndetachment event when the tip lost contact with this layer. Moreover, due to the ultrathin layer of\nmolten UHMWPE, this detachment \"jump\" during retraction is less sharp on the UHMWPE-\ncovered mica than on freshly cleaved mica.\nIf so, what is phase zero in their intermittent AFM? The baseline is not any more the real\nbaseline.\nWe thank the reviewer for raising this important question regarding the baseline in intermittent\ncontact AFM measurements. We appreciate the opportunity to present further details, which help\nto clarify our approach.\nIn our analysis, as the absolute phase baseline may vary between scan lines due to the exact\nabsolute amplitude of the cantilever oscillation influenced by thermal drift or environmental noise,\nthe value of the phase signal representing the mica substrate surrounding our UHMWPE filaments\nwas used as a reference, specifically for flattening the AFM phase images. It is also important to\nnote that taking the region of the mica substrate surrounding our UHMWPE filaments as the\nreference surface, this region may include additional interfacial layers, such as a thin layer of\nmolten UHMWPE melt, contributions from the water meniscus formed under ambient conditions,\nor even potential tip contamination from the UHMWPE melt. Nevertheless, as we always used the\nsame referencing procedure under these conditions, our interpretation remains both\nmethodologically consistent and physically justified, in particular as the overall response of the\nphase signal of this composite reference system remains stable and invariant over time. Crucially,\nany systematic shift in the reference value does not affect our interpretation of the presented phase\ncontrasts relative to this reference value. The clear distinction in phase values between the filament\nand the mica substrate surrounding our UHMWPE filaments, as well as the contrast between the\ntwo distinct regions within the filament itself, remains unequivocally discernible regardless of\nbaseline adjustments.\nInformation.\nKnowing that both, a water meniscus and an ultra-thin polyethylene (PE) melt layer cover the\nentire UHMWPE sample, a critical question arises: whether the phase value from the mica\nsubstrate surrounding our UHMWPE filaments can still serve as a reliable reference for flattening\nthe AFM phase data, and, if this flattening is necessary?\nAs the absolute phase baseline may vary between scan lines due to factors such as thermal drift or\nenvironmental fluctuations, the value of the phase signal representing the mica substrate\nsurrounding our UHMWPE filaments was used as a reference, specifically for flattening the AFM\nphase images. It is also important to note that taking the region of the mica substrate surrounding\nour UHMWPE filaments as the reference surface, this region may include additional interfacial\nlayers, such as a thin layer of molten UHMWPE melt, contributions from the water meniscus\nformed under ambient conditions, or even potential tip contamination from the UHMWPE melt.\nNevertheless, as we always used the same referencing procedure under these conditions, our\ninterpretation remains both methodologically consistent and physically justified, in particular as\nthe overall response of the phase signal of this composite reference system remains stable and\ninvariant over time. Crucially, any systematic shift in the reference value does not affect our\ninterpretation of the presented phase contrasts relative to this reference value. The clear distinction\nin phase values between the filament and the mica substrate surrounding our UHMWPE filaments,\nas well as the contrast between the two distinct regions within the filament itself, remains\nunequivocally discernible regardless of baseline adjustments.\nFurthermore, these trends are also observed event at the top of “hard” filaments. How they\ninterpret it? Every discussion made in this manuscript need to be modified.\nWe sincerely thank the reviewer for this insightful observation and for requesting further\nclarification. Similar to the other regions of the examined filaments, the presence of the \"nearly-\nconstant force region\" and the \"extended tail region\" atop the \"hard\" filaments can be attributed to\nseveral factors: the formation of a water meniscus between the sample surface and the cantilever\ntip under ambient conditions, possible tip contamination from the UHMWPE melt, or the presence\n9\nof a thin layer of UHMWPE melt. As we discuss later in this reply in the context of “non-melting\ncrystalline surfaces”, we anticipate that the top of the hard filaments is either a non-melting surface\nwhich is not covered by any liquid layer or covered with an extremely thin layer of UHMWPE\nmelt which, under the conditions of our AFM measurements, is easily penetrated by the AFM tip.\nAlthough the figure is not in the main text and therefore this is just a comment, but the vertical\naxis is somewhat strange. When I converted the force value into deflection value for curve 10\n(assume that the spring constant is 5 N/m), the deflection difference between D of 5 nm and 10\nnm is more than 5 nm. Thus, the sensitivity calibration was wrongly performed.\nWe appreciate the reviewer’s careful attention to the calibration details. In response to the\ncomment regarding the vertical axis and inverse optical sensitivity calibration, we would like to\nclarify that our calibration procedure yields values that are physically reasonable. For instance,\ntaking the same curve 10 as an example and assuming a spring constant of 5 N/m, the\ncorresponding force change measured over 5 nm displacement (in Figure R1.1-2, this corresponds\nto the range from 10 nm to 5 nm in distance) yields approximately 25 nN, as illustrated in Figure\nR1.1-2. This result aligns well with theoretical expectations and supports the validity of our\ncalibration approach.\nFigure R1.1-2: For the calibration of the spring constant of the cantilever\nSI-7, soft layers are not symmetric especially when the scan direction is perpendicular to the\ndirection of the filament. This is surely due to the tip-shape effect. I know this portion is not so\neasy, but the authors should mention something about the effect. If possible, they may be able\nto deconvolute the tip effect and once again discuss the value of 𝑊 .\nsoft\nWe sincerely thank the reviewer for the highly positive feedback regarding the discussion of Fig.\n2, as well as for raising the important point concerning the asymmetry in the width observed in the\nsoft layers in Supplementary Figs. SI-4 and SI-7, particularly when scanning perpendicular to the\nfilament axis. We agree that this asymmetry is likely influenced by tip-shape effects and scanning\norientation as well as scanning direction relative to the height gradient of the sample.\nIn response to this comment, in Figure R1.1-3, we have included both trace and retrace phase\nimages for two representative filaments. The observed asymmetries are mainly related to the\ndirection of scanning over the sample topography. Specifically, when scanning perpendicular to\nthe filament, the tip encounters an ascending and then descending slope, resulting in phase profiles\nin the trace and retrace directions that exhibit “mirror-image asymmetry” — as clearly\ndemonstrated in Figure R1.1-3.\nTo minimize potential errors introduced by this effect, all values of 𝑊 reported in the\nsoft\nmanuscript were measured from cross-sectional phase profiles obtained when the cantilever was\nscanning in the direction of ascending the soft layer slope. For instance, as illustrated in Figure\nR1.1-3, the width of the left soft layer was measured from the trace scan (where the tip ascends\nthe left slope), while the right soft layer was measured from the retrace scan (where the tip ascends\nthe right slope). This approach ensured consistent and reliable quantification of the soft layer width,\nreducing artifacts related to scan direction.\nWe acknowledge that tip deconvolution represents a valuable future direction for refining our\nanalysis. We thank the reviewer for highlighting this possibility for further methodological\ndevelopment. However, for the central message of our manuscript concerning the change in width\nof the soft layer with temperature, such a deconvolution does not seem to be required.\nasymmetry is a result of whether the AFM tip is scanning in trace or retrace direction. Trace (a)\nand retrace (c) AFM phase images taken from the same filament (taken in a single AFM\nmeasurement under ambient conditions), similar to the conditions used for obtaining results shown\nin Fig. SI-7. Both images have the same color code and scale bar. Trace (b) and retrace (d) AFM\nphase images taken from the same filament (taken in one AFM measurement under ambient\nconditions), similar to the condition used for obtaining results shown in Fig. SI-4. Both images\nhave the same color code and scale bar. The scan direction used for obtaining these images is\nlabeled on their top (bottom) with a red arrow.\n8. Dependence of the AFM phase signal on the scanning orientation relative to the height\ngradient of the sample.\nFig. SI-11: AFM phase trace and corresponding retrace images indicate that the observed\nasymmetry is a result of whether the AFM tip is scanning in trace or retrace direction. Trace (a)\nand retrace (c) AFM phase images taken from the same filament (taken in a single AFM\nmeasurement under ambient conditions), similar to the condition of Fig. SI-8. Both images have\nthe same color code and scale bar. Trace (b) and retrace (d) AFM phase images taken from the\nsame filament (taken in one AFM measurement under ambient conditions), similar to the condition\nof Fig. SI-4. Both images have the same color code and scale bar. The scan direction of those\nimages is labeled on their top (bottom) with a red arrow.\nThe asymmetry observed in the soft layers in Supplementary Figs. SI-4 and SI-8, particularly when\nscanning perpendicular to the filament axis, are likely influenced by the scanning orientation\nrelative to the height gradient of the sample.\nfilaments. The observed asymmetries are mainly related to the direction of scanning over the\nsample topography. Specifically, when scanning perpendicular to the filament, the tip encounters\nan ascending and then descending slope, resulting in phase profiles in the trace and retrace\ndirections that exhibit “mirror-image asymmetry” — as clearly demonstrated in Fig. SI-11.\nTo minimize potential errors introduced by this effect, all values of 𝑊 reported in the\nsoft\nmanuscript were measured from cross-sectional phase profiles obtained when the cantilever was\nscanning in the direction of ascending the soft layer slope. For instance, as illustrated in Fig. SI-\n11, the width of the left soft layer was measured from the trace scan (where the tip ascends the left\nslope), while the right soft layer was measured from the retrace scan (where the tip ascends the\nright slope). This approach ensured consistent and reliable quantification of the soft layer width,\nreducing artifacts related to scan direction.\nFinally, this comment is not so critical but they should clearly describe the resonant frequency\nof the cantilever and actual vibration frequency. They may better to describe how to measure\nthe resonant frequency. Just very far away from the sample surface like written in the manual\nof AFM or perform it at the vicinity of surface? I am asking this question because they used\nvery small amplitude ratio. The resonant frequency changes a lot due to long-range interaction\nforce or air damping. If they were not so careful, all the measurement is not in “soft tapping”\nregion but rather strong force is exerted on the specimen.\nWe thank the reviewer for raising this pertinent point regarding the characterization of the\ncantilever’s resonance frequency and vibration parameters, which are essential to ensure that\nmeasurements were conducted within the soft tapping regime. We fully agree that careful\ncalibration is crucial to minimize the influence of long-range forces and environmental factors.\nIn our experiments, the resonance frequency of the cantilever (approximately 165 kHz) was\ninitially determined at a minimum distance of 100 μm away from the sample surface to avoid any\n(also long range) tip–sample interactions. To account for initial instrumental instabilities like\nthermal drifts, all data collected within the first hour of operation were excluded from our study.\nImmediately before acquiring the data presented in the manuscript and in the Supplementary\nthe same non-interactive conditions (at a distance ≥100 μm above the surface). Furthermore, the\nfree vibration amplitude was verified after each measurement using the procedure detailed in Fig.\nSI-9(a), where the plateau region of the amplitude-distance curve provides a direct measure of this\nparameter. This systematic approach ensured that the operating parameters remained within the\nsoft tapping regime throughout the measurements, thereby minimizing the force exerted on the\nsample.\nIn the \"Methods- Characterization techniques\" section of our manuscript, “a characteristic spring\nconstant of the cantilever of around 48 N/m, and a resonance frequency of the cantilever of around\n165k Hz” has been added to the first paragraph.\nIn the \"Methods- Characterization techniques\" section of our manuscript, “obtained at a position\nwhere the cantilever tip was at least 100 μm from the sample surface” has been added to the third\nparagraph, the information “the AFM cantilever oscillates at a frequency slightly lower (in the\norder of 100 Hz) than one of the resonance frequencies” has been added. The full sentence now\nreads as: “During a typical intermittent-contact mode AFM measurement (in air), the AFM\ncantilever oscillates at a frequency slightly lower (in the order of 100 Hz) than one of the resonance\nfrequencies with a \"free\" amplitude 𝐴 , obtained at a position where the cantilever tip was at least\n0\n100 μm from the sample surface.”\nThis paper provides convincing evidence for ~9-nm thin liquid layers along the edges of mica-\nsupported semicylindrical polyethylene filaments, obtained by AFM, and tries to explain their\ntemperature dependence. This is an intriguing system, which was apparently introduced earlier\n(see ref. 57 from 2015).\nWe are deeply grateful to the reviewer for the overall positive assessment of our work. We truly\nappreciate the recognition of the convincing evidence for the ~9-nm width liquid layers along the\nedges of the mica-supported semi-cylindrical polyethylene filaments obtained by AFM, as well as\nthe interest in our explanation of the temperature dependence of these structures. It is particularly\nencouraging to us that the reviewer finds our studied system intriguing, and we are thankful for\nacknowledging its connection to some of our earlier studies. In ref. 57, polymer filaments of\nseveral micrometers were generated by a dewetting process. In the present manuscript, we used\nspin coating under different conditions, allowing that polymer chains undergo a coil-to-stretch\ntransition and forming ultra-long filaments of up to millimeters. We would like to emphasize that\nwe anticipate that most polymer chains were (almost) fully stretched, reducing the number of\npossible chain folds and chain entanglements within the filaments drastically. Thus, the thin liquid\nlayers along the edges of mica-supported semi-cylindrical polyethylene filaments are expected to\nbe in contact with a crystalline surface. This is in contrast to a possible liquid-like layer on the fold\nsurface of a lamellar polymer crystal and is also different from a so-called “rigid-amorphous\nfaction”, mainly located at the fold surface of polymer crystals.\nThe title “Temperature dependence of crystal – melt coexistence for supported polyethylene\nfilaments” does not describe the content of the paper accurately. (Below I’m proposing some\nexperiments that could help fulfill the promise of this more ambitious title.) As it stands,\n“Temperature dependence of liquid edge layers of supported polyethylene filaments” would be\nan accurate title.\nWe sincerely appreciate the reviewer’s thoughtful suggestion regarding the title of our manuscript.\nWe agree that clarity and accuracy are essential. The proposed alternative title, “Temperature\nDependence of Liquid Edge Layers of Supported Polyethylene Filaments,” certainly captures our\n16\nkey experimental observation. At the same time, we believe that the original title, “Temperature\nDependence of Crystal–Melt Coexistence for Supported Polyethylene Filaments,” is better suited\nbecause it captures in addition the essential physical phenomenon underlying the formation of the\nliquid edge layers—namely, the equilibrium coexistence between crystalline and molten phases\ncharacterized by a length scale which depends on temperature.\nThe theory presented centrally in this paper requires major revisions. Already in the Abstract,\nthe authors name-drop “nucleation, surface premelting, and the melting-point depression of\nfinite-size”, which are later confirmed to refer to the critical nucleus in classical nucleation\ntheory (a lower limit of crystal size), premelting, and Gibbs-Thompson theory (another more\nstringent lower limit of crystal size). While at the end of Abstract and elsewhere, the authors\nclaim to have unified these phenomena, the theory remains at a disappointingly low level, mostly\njust citing diverse sources (see Table 1 and the scant text below it).\nWe sincerely appreciate the reviewer’s critical comment and the opportunity to clarify the scope\nand intention of our discussion of related theoretical concepts. We thank the reviewer for the\ncareful reading and acknowledge that our brief and limited presentation of the theoretical context\nmay have caused misunderstandings.\nWe agree that in our manuscript we only indicated/suggested the possibility of unifying various\nconcepts, i.e., the concepts of nucleation, surface premelting, and finite-size melting point\ndepression. Of course, a thorough proof of such unification requires a profound theoretical\ndevelopment, which is beyond the scope of this study. The primary goal of our work presented in\nthe manuscript was to provide a possible interpretation of the observed crystal–melt coexistence\nin polyethylene filaments. Thus, we have compared our results with predictions of the Lauritzen–\nHoffman theory of secondary nucleation at the growth front of the crystal. References to classical\nnucleation theory, surface premelting, and the Gibbs–Thomson effect were included to indicate\nthat these phenomena are related and can, within the uncertainty of our derived results on the\nwidths of the soft region, also explain our observations.\na liquid layer studied here.\nWe thank the reviewer for raising this important point regarding the applicability of classical\nnucleation theory (CNT) and the Gibbs–Thomson equation to interpret the thickness of the liquid\nlayer along the edges of mica-supported semi-cylindrical polyethylene filaments observed in our\nstudy. This comment allows us to provide more details on the theoretical links between a stringent\nlower limit of crystal size and the equilibrium thickness of the liquid layer (or, e.g., in the context\nof ice and water, also called quasi-liquid layer) on the surface of a crystal, as discussed and\npredicted in concepts of surface melting or pre-melting. As described in a large number of\npublications and also demonstrated in recent computer simulations on nanoscopic systems, in\nequilibrium, a crystal can be covered by a liquid layer of its melt. As shown, for example, in ref\n(1) of our manuscript, U. Tartaglino, T. Zykova-Timan, F. Ercolessi, E. Tosatti, Melting and\nnonmelting of solid surfaces and nanosystems, Phys. Rep. 411 (2005) 291–321,\nhttps://doi.org/10.1016/j.physrep.2005.01.004 , the thickness ℓ(𝑇) of this liquid layer can be\nrelated to the wetting behavior described via the corresponding surface and interfacial tensions.\nFor curved interfaces, one has to include additional terms like, for example, a term characterized\nby the Tolman length (see, e.g., P. Montero de Hijes, Jorge. R. Espinosa, Valentino Bianco,\nEduardo Sanz, and Carlos Vega, Interfacial Free Energy and Tolman Length of Curved\nLiquid-Solid Interfaces from Equilibrium Studies, J. Phys. Chem. C, 124, 8795-8805 (2020);\nhttps://doi.org/10.1021/acs.jpcc.0c00816 ). Represented through an “interaction” free energy\nterm 𝑉(ℓ), which includes the various short- and long-range intermolecular forces at work, the\nsolid–liquid and the liquid–vapor interfaces “feel” their mutual presence. As described through\nequation (2) in this above-mentioned paper by Tartaglino et al., the thickness ℓ(𝑇) of this layer\ndiverges as the melting temperature is approached. Thus, a crystal of finite size 𝜎 will be molten\ncompletely when the molten layer covers the whole size of the crystal., i.e., ℓ(𝑇) ≥ 𝜎. Consequently,\nsimilar to what we showed in Fig. 5 of our manuscript, the value of ℓ(𝑇) decides also the melting\ntemperature of finite size crystals.\nWe agree that it may not be obvious how nucleation and surface melting are related. The\nunderlying assumption is that nucleation is not possible when the equilibrium size of the coexisting\nliquid phase in a finite system is equal or even larger than the size 𝜎 of the whole system. As shown\n18\nin refs. (14) - (16) cited in our manuscript (and, for convenience, at the end of this paragraph), for\na fixed small number 𝑁 of molecules and a finite volume 𝑉 (i.e., a nanoscopic system) at a\ntemperature 𝑇, it is possible to observe stable interfaces between a crystalline and a liquid phase\nthat are in thermodynamic equilibrium. In these works, it has also been shown that this equilibrium\nsize of the crystalline phase in finite systems can be related to the size of a critical nucleus in large\n(infinite) systems, where crystal growth can be observed. The temperature-dependence of the size\nof the critical nucleus in infinite systems is used for the comparison of our experimental results on\nthe width 𝑊 of the liquid layers along the edges of mica-supported semi-cylindrical\nsoft\npolyethylene filament of finite size.\n(14) Binder, K.; Block, B. J.; Virnau, P.; Tröster, A. Beyond the Van Der Waals Loop: What\nCan Be Learned from Simulating Lennard-Jones Fluids inside the Region of Phase\nCoexistence. American Journal of Physics 2012, 80 (12), 1099–1109.\nhttps://doi.org/10.1119/1.4754020.\n(15): Statt, A.; Virnau, P.; Binder, K. Finite-Size Effects on Liquid-Solid Phase Coexistence\nand the Estimation of Crystal Nucleation Barriers. Phys. Rev. Lett. 2015, 114 (2), 026101.\nhttps://doi.org/10.1103/PhysRevLett.114.026101.\n(16) Montero de Hijes, P.; Vega, C. On the Thermodynamics of Curved Interfaces and the\nNucleation of Hard Spheres in a Finite System. The Journal of Chemical Physics 2022, 156 (1),\n014505. https://doi.org/10.1063/5.0072175.\nSpecifically, in the central Figure 5c, the authors use the two limiting crystal thicknesses to fit\nthe T-dependence of a liquid layer. In the caption of Table 1, they make the misleading claim\nthat the table shows “the characteristic length scale, represented as 𝑙, of a liquid layer on the\nsurface of a crystal, as predicted by three different phenomena”, while at least two of the entries\nare actually the limiting thicknesses of crystals.\nWe thank the reviewer for this insightful comment regarding the interpretation of characteristic\nlength scales in Figure 5c and Table 1. We agree that the caption may have been misleading in its\npresentation. We have thus improved its clarity.\na crystalline and a liquid phase in a finite system, the underlying assumption is that nucleation is\nonly possible when the equilibrium size of the coexisting liquid phase in a finite system is less\nthan the whole system. Thus, if the volume fraction of the liquid phase increases, the volume\nfraction of the solid phase decreases until it disappears at the melting temperature of this finite size\nsystem. Of course, geometry of the system of finite size matters but also the geometry of the (often\nfaceted) solid phase is relevant. Thus, it can only be an approximation when we represent the\nrespective volumes by a single “characteristic” length scale. For most theoretical concepts we used\na thickness ℓ(𝑇) while for our experimental results we represent this length scale by the width\n𝑊 of the liquid layers along the edges of mica-supported semi-cylindrical polyethylene\nsoft\nfilament of finite size. When describing the volume fraction of the solid phase, we mainly assume\nthat it has a cylindrical shape, allowing a description by its radius 𝑟 (or 𝑟∗ when we consider a\ncritical nucleus). As shown, for example, in refs. (14) - (16) cited in our manuscript, in most cases\nthe geometry and the curvature of the system or that of the solid phase can be accounted for by an\nappropriately derived prefactor. Thus, describing the phases through a single length scale is a\nuseful approximation, similar to the concept of the Tolman length scale which was introduced to\naccount for the impact of curved interfaces on nucleation via changes in interfacial energies\nquantified through a length scale, i.e., the Tolman length.\nIn our manuscript, we do not intend to merge distinctly different concepts. However, we want to\nillustrate that the same (mean) intermolecular forces acting at interfaces, which vary with the\ndensity of solid and liquid phases, are responsible for different phenomena, determining\ncharacteristic features of the coexistence of a crystal with its melt. Thus, representing this\ncoexistence by a characteristic length scale allows us to compare, at least qualitatively, the\npredicted temperature dependence of the various concepts.\nOn p. 2, we read “thickness l of the wetting layer” while the Abstract described “molten regions\nof rather uniform width, Wsoft” - how are l and Wsoft different? It seems that generic l is\nintroduced to hide the fact that sometimes it refers to a crystal thickness and sometimes to the\nthickness of a liquid wetting layer. If in the table instead of “l ~” the authors wrote “Wsoft ~”\nfollowed by a crystallite thickness, the inconsistency would become apparent. On p. 13, r* is\nconfusingly added to the mix of symbols, but it is treated as l in Table 1.\ndistinction in our terminology. The parameter \" ℓ \" in our manuscript refers to the theoretically\npredicted thickness of a molten polymer wetting layer on an idealized, smooth crystalline substrate,\nas derived from various thermodynamic models. In contrast, \"𝑊 \" represents the experimentally\nsoft\nmeasured width of the soft, molten PE meniscus layer formed between the PE nanofilament and\nthe mica substrate, derived from our AFM phase data. In our answers to the following comments,\nwe will clarify further the meaning and relations between the various length scales introduced and\ndiscussed in our manuscript.\nThe theory on p. 13 contains technical flaws in the definition of the quantities (see details below),\nwhich suggests that the authors were struggling with the concepts. It seems that the simple\ntheory on p. 13 is suitable for primary nucleation, but the authors make it clear that they want\nto consider secondary nucleation. In secondary nucleation, there should be two characteristic\nlength scales, along the preexisting surface and perpendicular to it, not just a single r*. I see\nthis confirmed in the corresponding author’s cartoon of a secondary critical nucleus in\nreference 50, which is one chain stem thick and five stems wide. The assumption of a cylindrical\nsecondary nucleus (top of p. 12) is unreasonable.\nWe sincerely thank the reviewer for this insightful comment and for providing the opportunity to\nclarify our theoretical approach. We appreciate the careful reading and acknowledge the need for\nmore detailed descriptions for the CNT.\nIn the initial version of the manuscript, the critical size of the secondary nucleus was modeled as\na cylinder with volume 𝑉 = 𝜋𝑟∗2 ∙𝑙∗. There, two characteristic length scales are used: “𝑟∗” is the\ncr\nradius and “𝑙∗” is the length of the cylinder. Given that the polymer chains within the nanofilament\nare oriented along its axis, and considering that the orientation of secondary nuclei is governed by\nthe crystalline face of original crystal—which is aligned along the nanofilament—we consequently\nassume that the polymer chains inside the secondary nuclei are likewise aligned parallel to their\nrespective nanofilaments. As a result, “𝑙∗” is oriented parallel to the nanofilament, while “𝑟∗” is\nperpendicular to it. It is therefore “𝑟∗” that governs the value of \"𝑊 \", and for this reason, our\nsoft\nsubsequent analysis focuses specifically on this parameter.\nmultiple length scales. In reference [50] and also in I Liu, Y., Wang, Z., Zhang, Y. et al. The size\nof critical secondary nuclei of polymer crystals does not depend on supersaturation. Nat\nCommun 16, 3773 (2025). https://doi.org/10.1038/s41467-025-58962-5, we have given some\nexamples. We appreciate the reviewer’s emphasis on this point. Accordingly, we revised the\nrelevant section of the manuscript to clearly justify the use of a cylindrical representation,\nexplicitly define both dimensions, and explain the physical reasoning why we have focused only\non “𝑟∗” in the context of our experimental system.\nThe sentence “Moreover, since the secondary nuclei adopt the same orientation as the original\ncrystal—that is, the nanofilament in which the polymer chains are aligned along its axis—the\ncylindrical nucleus is aligned lengthwise along the nanofilament, while its radius is oriented\nperpendicularly; this radius is directly correlated with the value of 𝑊 , i.e., 𝑊 ~ 𝑟∗. This\nsoft soft\nimplies that the crystal-melt coexistence observed at the growth front of the polymer crystal differs\nfrom the well-studied coexistence at the fold surface.” has been added to page 12, after the sentence:\n“Accordingly, if the residual number of molten polymer chains at the growth front per appropriate\nlength along the filament adds up a total volume less than 𝑉 , these polymer chains cannot\ncr\ncrystallize and will stay in the molten state.”.\nAnd why can’t the existing fibrillar crystal grow without nucleation, as in Regime I of Lauritzen-\nHoffman theory?\nWe sincerely thank the reviewer for raising this fundamental question regarding the applicability\nof the Lauritzen-Hoffman theory to our system. We appreciate the opportunity to clarify the role\nof secondary nucleation in the growth of existing nanofilament crystals.\nIt is our understanding that even within Regime I of the Lauritzen-Hoffman theory, crystal growth\nproceeds via a secondary nucleation process, albeit at a relatively slow rate compared to the\nsubsequent substrate layer completion step. Like in many previous publications by multiple\nauthors, the concept of secondary nucleation and its relation to homogeneous (primary) nucleation\nhas been discussed also in a recent publication: Liu, Y., Wang, Z., Zhang, Y. et al. The size\nof critical secondary nuclei of polymer crystals does not depend on supersaturation. Nat\nCommun 16, 3773 (2025). https://doi.org/10.1038/s41467-025-58962-5 . The key difference\n22\nbetween Regime I and Regime II lies in the rate of secondary nucleation; in Regime I, the spreading\nand completion of the crystalline layer are much faster than the formation of a new secondary\nnucleus on the growth face, leading to nearly homogeneous growth in the lateral direction.\nHowever, such homogeneous growth does not imply the absence of secondary nucleation, but\nrather reflects a specific kinetic relationship between its rate and the rate of surface spreading of\nthe crystalline layer.\nWe have based our interpretation on the foundational work by Hoffman, Davis, and Lauritzen (as\ndetailed in Section 3.4.2 “Growth Rate for the Case Where the Formation of a Surface Nucleus\nIs Followed by Rapid Completion of the Substrate (Regime I)” of their chapter in Treatise on\nSolid State Chemistry, Springer, 1976), which establishes that secondary nucleation is an integral\ncomponent of the growth mechanism in both regimes for chain-folded polymer crystals.\nThe formation of fibrillar crystals follows the same concepts of nucleation. As discussed in our\nprevious publication *, even stretched polymer chains, which are attracted via van der Waals\ninteractions, need to form “bundles” of a certain (critical) size in order to become crystalline.\n* ref. (28) in our manuscript:\nDa Huang, Thorsten Hugel, and Günter Reiter, Mechanism for the Formation of Millimeter-\nLong Solid Filaments by Spin Coating Dilute Solutions of a Crystallizable Polymer,\nMacromolecules 2024 57 (8), 3696-3705; https://doi.org/10.1021/acs.macromol.3c02524\nWhat the data do seem to show is premelting. It is included in Table 1 but not really discussed.\nIt needs to be explained that while the crystal is the lowest free-energy form of the polymer,\nsurface energy of the air-polymer interface makes insertion of a liquid-like layer\nthermodynamically favorable. It should also be discussed that in bulk polymers, premelting has\nbeen invoked, e.g. by Tanabe, Strobl, and Fischer (https://doi.org/10.1016/0032-3861(86)90001-\n7) in studies of HDPE, with a different meaning: without air, at the amorphous-crystalline\ninterface, eating away part of the crystal as temperature is raised, quite analogous to the\nobservations in this manuscript – why is this favorable thermodynamically?\ncrystal-melt coexistence of supported PE nanofilaments. As a first interpretation of our\nexperimental observations described through 𝑊 (𝑇), we have chosen CNT, employed within\nsoft\nthe concept of secondary nucleation. There, the underlying assumption is that nucleation is only\npossible if enough molten molecules are available to form a nucleus of critical size, approximately\ndescribed through the parameter 𝑟∗. This assumption leads to the condition that no nuclei are\nformed inside the soft meniscus layer of molten PE, which is characterized by 𝑊 (𝑇), when\nsoft\n𝑊 (𝑇)~𝑟∗(𝑇), due to the limitation of the size of the meniscus, i.e., the lack of a sufficient\nsoft\nnumber of molten molecules. As predicted by theory on surface melting or premelting and\nconfirmed by various experimental systems, a crystalline surface may be covered by a liquid layer\nof molten molecules of the same kind. We related the characteristic length scale of this predicted\nliquid layer with our experimental results on the width 𝑊 of the liquid layers along the edges\nsoft\nof mica-supported semi-cylindrical polyethylene filament of finite size (width). There, the\nunderlying assumption is that a filament where all polymer chains would in a crystalline state at a\ntemperature of 𝑇 = 0 K has to be covered by molten polymer chains at 𝑇 ≫ 0 K. The relation\nof 𝑊 (𝑇) with the reduced melting temperature of crystals of finite size 𝜎 invokes that no\nsoft\ncrystalline phase is remaining if the width of the liquid layer of molten molecules covers the whole\nfinite system.\nAs discussed in the above section, polymer chains within the nanofilament are almost exclusively\noriented parallel to the filament axis. There is no surface covered with chain folds as it is exhibited\nin lamellar crystals of chain-folded polymers. Consequently, in the case of the investigated\nfilaments, molten polymer chains are coexisting on the lateral surface of the crystalline core of the\nfilaments, which may be identified as the \"growth front\" of the “oriented” polyethylene crystal,\nrather than the molten phase on the fold surface of lamellar polymer crystals, as reported by\nnumerous other researchers, e.g., also by Tanabe, Strobl, and Fischer, Surface melting in melt-\ncrystallized linear polyethylene, Polymer, 27, 1147-1153 (1986); https://doi.org/10.1016/0032-\n3861(86)90001-7 .\nWe note that all concepts discussed above are based on thermodynamic parameters which apply\nto equilibrium situations. None of these concepts include any kinetic (time-dependent) aspects.\nOur results on 𝑊 (𝑇) show no dependence on the age of the sample, do not depend on filament\nsoft\n24\nsize or morphology, and have been shown do be independent of molecular weight and dispersity\nof the polyethylene used. The interpretation of the results on surface melting by Tanabe et al. and\nlater versions by M. Muthukumar† or by J.U. Sommer‡ also rely on equilibrium, i.e.,\nthermodynamic concepts.\n† Muthukumar, M. (2007). Shifting Paradigms in Polymer Crystallization. In: Reiter, G., Strobl,\nG.R. (eds) Progress in Understanding of Polymer Crystallization. Lecture Notes in Physics,\nvol 714. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-47307-6_1\n‡ Sommer, JU. (2007). Theoretical Aspects of the Equilibrium State of Chain Crystals. In:\nReiter, G., Strobl, G.R. (eds) Progress in Understanding of Polymer Crystallization. Lecture\nNotes in Physics, vol 714. Springer, Berlin, Heidelberg.https://doi.org/10.1007/3-540-47307-6_2\nThe literature on conventional premelting tells us that any crystal, in particular near its melting\npoint, is covered with a liquid-like layer. Why is there no such liquid layer on top of the\ncrystalline filament in Figure 3?\nWe sincerely thank the reviewer for raising this insightful question regarding the apparent absence\nof a liquid-like layer on top of the crystalline filament in Figure 3, which helps us clarify an\nimportant aspect of our experimental observations.\nAs described in theoretical treatments (see, e.g., ref (1) of our manuscript, U. Tartaglino, T.\nZykova-Timan, F. Ercolessi, E. Tosatti, Melting and nonmelting of solid surfaces and\nnanosystems, Phys. Rep. 411 (2005) 291–321, https://doi.org/10.1016/j.physrep.2005.01.004 ),\nand demonstrated in experiments (see, e.g., ref (4) of our manuscript, Pluis, B.; van der Gon, A.\nW. D.; Frenken, J. W. M.; van der Veen, J. F. Crystal-Face Dependence of Surface Melting.\nPhys. Rev. Lett., 59, 2678 (1987). https://doi.org/10.1103/PhysRevLett.59.2678), depending on\nthe interfacial properties, a thin molten layer could indeed form on the top surface of the crystalline\nnanofilament. However, besides surface melting, there exist also conditions for non-melting of\nsolid surfaces. Even within a single crystal (see, for example, lead), some crystal faces may exhibit\nsurface melting but other faces are not covered by a liquid layer (see, for example, D. Schebarchov\nand S. C. Hendy, Superheating and Solid-Liquid Phase Coexistence in Nanoparticles with\nNonmelting Surfaces, Phys. Rev. Lett. 96, 256101 (2006);\nhttps://doi.org/10.1103/PhysRevLett.96.256101 ), leading to the curious observation of\nsuperheating of such crystalline faces.\nsubstrate, we have to deal with different values of the interfacial energies and surface tensions\nacross the filament. Thus, we may anticipate that not all parts of the filament are covered with a\nliquid layer and the conditions for surface melting (or non-melting) are not fulfilled for all parts of\nthe nanofilament. Thus, probably due to the opposite curvature at the top of the filament compared\nto the region along the edges of mica-supported semi-cylindrical polyethylene filaments, we\nobserved distinctly different wetting conditions, leading to the strong contrast in the thickness of\nthe layer of molten polymers. The layer at the top of the filament, if it exists at all, is expected to\nbe substantially thinner than in the meniscus region formed at the edges where the filament meets\nthe mica substrate.\nConsequently, the AFM phase signal measured on top of the nanofilament is dominated by the\nmechanical response of the underlying crystalline material, possibly with a minor (insignificant)\ncontribution from an ultrathin surface liquid layer. In contrast, the meniscus region exhibits a much\ngreater effective thickness. There, the AFM phase response is clearly dominated by the properties\nof the molten polyethylene—at least until the tip fully penetrates this region. This pronounced\ndifference in liquid layer thickness is reflected in the strong phase contrast between the soft\nmeniscus and the rigid crystalline filament, enabling robust and reproducible measurement of the\nmeniscus width, which constitutes the focus of our analysis.\nWe appreciate the reviewer’s comment and incorporated this clarification in the revised\nmanuscript to better articulate the geometric and detection-related reasons behind the observed\ncontrast.\nThe sentence \"Due to the semi-cylindrical geometry of the nanofilament and its interaction with\nthe wettable mica substrate, we have to deal with different values of the interfacial energies and\nsurface tensions across the filament. One possibility is that not all parts of the filament are covered\nwith a liquid layer and the conditions for surface melting (or non-melting) are not fulfilled for all\nparts of the nanofilament. The other possibility is that an ultrathin liquid layer exists at the top of\nthe filament. In the latter case, it is so thin that its AFM phase signal is dominated by the\nmechanical response of the underlying crystalline material, possibly with a minor (insignificant)\ncontribution from this ultrathin surface liquid layer. In contrast, the meniscus region exhibits a\n26\nmuch greater effective thickness. There, the AFM phase response is clearly dominated by the\nproperties of the molten polyethylene—at least until the tip fully penetrates this region. Thus, the\nobserved soft regions in the AFM phase images correspond to the molten polymer chains in the\nconcave meniscus region at the edges.\" has been added to the paragraph after Figure 3, after the\nsecond sentence.\nOne could envision that entanglements that have been excluded from the growing crystal and\nenriched outside the crystal prevent crystallization of the liquid layer. This is invoked by some\nexperts to explain lamellar crystallites alternating with amorphous layers in semicrystalline\npolymers. Is this entanglement mechanism excluded for the liquid layers observed in this study?\nAs discussed in our ref. (28) [Huang, D.; Hugel, T.; Reiter, G. Mechanism for the Formation of\nMillimeter-Long Solid Filaments by Spin Coating Dilute Solutions of a Crystallizable\nPolymer. Macromolecules 2024, 57 (8), 3696–3705. https://doi.org/10.1021/acs.macromol.\n3c02524.], the polymer chains within the nanofilament are highly stretched and preferentially\noriented parallel to the filament axis. In the course of filament preparation via spin coating, even\nself-entanglements are removed efficiently. To induce filament formation, we require that the\nhighly stretched chains, oriented and aligned in the shear flow direction, attract each other. Thus,\nwe do not expect that there is any significant amount of entanglements between polymer chains\nwithin the filaments. Consequently, the crystal-melt coexistence phenomenon described in our\nmanuscript occurs specifically at the lateral crystalline \"growth front\" of the polyethylene\nfilaments. Due to these distinct features of the polymer chains, which mostly underwent a coil-to-\nstretch transition, the crystal-melt coexistence behavior is not affected by any impact of\nentanglements. Thus, our crystalline polymer structures are different to, for example, stacks of\nchain-folded lamellar polymer crystals.\nOn p. 2, at the very beginning of the Introduction, the authors do not present a motivation for\ntheir work. I do find it intrinsically interesting, but it would be better if a motivation or grander\nframework was included here.\nWe sincerely thank the reviewer for this constructive suggestion. We fully agree that providing a\nwide-ranging motivation at the beginning of the Introduction will significantly improve the\n27\nmanuscript by framing our work within a broader scientific context. Thus, we have revised the\nopening paragraph to articulate the fundamental questions and the broader implications of studying\ncrystal-melt coexistence, the critical size required for (secondary) nucleation and its relation to\nsurface melting and the melting point reduction in nanoscopic polymer filaments. We hope that\nsuch a more detailed and comprehensive description of the main objectives ensures that the\nsignificance of our study becomes clear to all readers from the outset.\nOn page 3, the first sentence of the second paragraph has been rephrased as \"In the study presented\nhere, we experimentally explored the coexistence of a nanoscopic polymer crystal with a liquid\n(molten) layer characterized by a thickness, which we compared with predictions of the Lauritzen–\nHoffman theory of secondary nucleation at the crystal growth front, represented by the\ncomplementary length scale of the critical nucleus. Our findings suggest that, for nanoscopic\nsystems, the temperature dependence of the characteristic length scales of various phenomena, i.e.,\nthe size of the critical nucleus, the thickness of the liquid layer in surface premelting, and the size-\ndependence of the melting-point depression of finite-size crystals are related.\".\nFigure 5c displays intriguing images that raise interesting questions. The image at top shows a\nvery thin crystal (the bright region about 7 nm across) at high temperature – is such a small\nthickness compatible with the Gibbs-Thompson equation, which tells us that such thin crystals melt\naway, in particular at high T? Have the authors raised temperature and seen the crystal melt\nsuddenly when its thickness drops below the Gibbs-Thompson stability limit?\nAt the bottom of the figure, we see a completely melted filament. Observing it at gradually\ndecreasing temperature when at some point a critical nucleus must form would seem like a\nworthwhile investigation, but the authors do not seem to consider critical nuclei in this relevant\ncontext.\nWe sincerely thank the reviewer for raising these insightful questions regarding the stability of\nthin crystals and the potential for real-time observation of melting and nucleation events.\nAs mentioned by the reviewer, the Gibbs-Thomson equation tells us that thin crystals melt away,\nin particular at high temperatures. In Figure 5c, we have presented this prediction of the Gibbs-\nThomson equation (using a suitable prefactor) by the black line which is exactly telling that at high\n28\ntemperatures the filament width 𝑤 needs to be thicker than roughly two times 𝑊 . In the\nsoft\nmanuscript we wrote: “Filaments are completely molten for 𝑤 ≲ 2∙𝑊 (𝑇).”\nsoft\nFurthermore, also in Figure 5c, for a fixed temperature of 𝑇 ≈100 °C, we have shown that a\nthreshold value of the size of the crystal or of the filament size 𝑤 exists, as predicted by the Gibbs-\nThomason equation. In the insets of Figure 5c, we showed AFM phase images for two filaments\nwith 𝑤 ≈ 26 nm and 𝑤 ≈ 42 nm, respectively. In full agreement with the Gibbs-Thomson\nequation, the smaller filament was completely molten while the somewhat wider filament exhibit\nthe remains of a tiny solid region (the reviewer measured the width of this bright region to be about\n7 nm across).\nWe agree that ramping the temperature during a running AFM scan would be highly insightful for\nfundamental aspects of polymer crystallization. However, capturing melting of ultra-thin crystals\nupon heating or the formation of critical nuclei upon cooling is unfortunately not feasible with our\ncurrent experimental setup due to inherent technical limitations. A primary constraint is the\nsignificant thermal expansion of both the sample and the heating stage during temperature changes.\nThis expansion causes the sample surface to move vertically toward the AFM tip on the cantilever.\nSince our high-resolution imaging requires operating at the minimal Z-range limit of the\npiezoelectric scanner to maintain optimal sensitivity, even a minor amount of vertical displacement\neasily exceeds the available range, preventing stable imaging under these conditions.\nTaking all these technical limitations into consideration, we can still confirm that the thinner\nfilament with 𝑤 ≈ 26 nm was already molten when we reached the temperature of 𝑇 ≈100 °C\nand started the AFM measurement. Thus, within this admittedly large experimental time window,\nthe melting process occurred.\nFurthermore, as suggested by the reviewer, observing nucleation at gradually decreasing\ntemperature is definitely a worthwhile investigation, also because theoretical works like the\nseminal paper by Lipowsky [R. Lipowsky, Critical Surface Phenomena at First-Order Bulk\nTransitions, Phys. Rev. Lett., 49, 1575-1578 (1982),\nhttps://doi.org/10.1103/PhysRevLett.49.1575 ] predict that the surface order parameter may\nbehave continuously with temperature approaching the transition temperature, although the bulk\n29\norder parameter jumps at the transition temperature. Our experimental observations (considering\nall its inherent technical limitations, in particular with respect to its temporal resolution) suggest a\ncontinuous change in 𝑊 as temperature approaching the transition temperature. Of course, the\nsoft\nkey question raised by the reviewer on how nucleation occurs within a molten filament is still open.\nWe only can state that (again accounting for the long times required for eliminating drifts induced\nby temperature changes) the filament with w≈26 nm, which was completely molten at the\ntemperature of T ≈100 °C, showed a crystalline core and a value of 𝑊 ≈ 9 nm when cooled\nsoft\nback to room temperature, demonstrating that nucleation occurred when cooling the sample.\nI’m quite uncomfortable with the way in which the authors use the term “polymers” (which they\ndo a lot, 50 times in total). To me, examples of polymers are PE, iPP, and PET. The authors use\n“polymers” where I would prefer “polymer chains”, “polymer segments”, or “polymer”.\nExamples:\nP. 3, center: “the limited number of polymers available” should probably be changed to “the\nlimited number of polymer chains available”.\nP. 3, 5th line from bottom: “interactions between stretched polymers,” should probably be\nchanged to “interactions between stretched polymer chains,”.\nP. 4, first paragraph: “filaments of crystalline chains were surrounded initially by non-stretched\nmolten polymers.” should probably be changed to “filaments of crystalline chains were\nsurrounded initially by non-stretched molten polymer segments.”\nP. 4, 2nd paragraph: “the number of available polymers is above a threshold value.” Should be\nchanged to “the number of available polymer chains [or segments?] is above a threshold value.”\nP. 4, below center: “sufficient mobility of noncrystallized polymers but also” should be changed\nto “sufficient mobility of noncrystallized polymer but also” or “sufficient mobility of\nnoncrystallized polymer segments but also”\nP. 5, second sentence: “contained molten polymers only”\nP. 18, line 7/8: “contained molten polymers which bounded the solid core region of the filament\nconsisting of crystallized polymers.” should be changed to “contained molten polymer which\nbounded the solid core region of the filament consisting of crystallized polymer.”\nWe thank the reviewer for this precise and helpful comment regarding the use of the term\n\"polymers.\" We agree that using more specific terminology such as \"polymer chains\" will enhance\nthe clarity and precision of our descriptions. We have carefully revised the manuscript accordingly\nand replaced the general term \"polymers\" with the more accurate \"polymer chains\" throughout the\ntext, ensuring that no unintended ambiguity is introduced.\nFor corrections, two different changes are made. (1), \"polymer\" or \"polymers\" is replaced by\n\"polymer chain\" or \"polymer chains\" when it refers to an or many polymer chains; (2), \"polymers\"\nis replaced by \"polymer\" when it refers to a single kind of polymer. Positions where correctios\nwere applied are shown below:\nIn the following cases, the words \"polymer\" or \"polymers\" have been replaced by \"polymer chain\"\nor \"polymer chains\", respectively:\n P(age)3: To generate separate polymer crystals of small size, we required a system with a highly\nlimited number of polymer chains.\n P3: Previous experiments23 have shown that polymer chains in ultra-thin layers,\n P3: due to the limited number of polymer chains available in such ultra-thin films,\n P3: the homogeneous nucleation step may be circumvented by introducing filaments consisting\nof stretched and uniquely oriented polymer chains\n P3: a shear-induced change in polymer chain conformation can occur\n P3: i.e., polymer chains undergo a transition from randomly coiled to stretched chains.\n P3: attractive interactions between stretched polymer chains,\n P3: Due to the massive loss of conformational entropy and the high degree of order of the polymer\nchains within the filaments\n P4: polymer chains may crystallize easily within the filament\n31\n P4: Thus, not all polymer chains can undergo such a coil-stretch transition\n P4: filaments of stretched and crystalline chains were surrounded initially by non-stretched\nmolten polymer chains.\n P4: which requires that the number of available polymer chains is above a threshold value.\n P4: we anticipate that crystal growth will stop once the number of remaining molten polymer\nchains is below this threshold value.\n P4: allowing for sufficient mobility of non-crystallized polymer chains but also for crystal growth\n P4: Thus, all polymer chains had a chance to crystallize\n P5: If these filaments would have contained molten polymer chains only\n P9: Thus, as a consequence of the acting capillary forces, polymer chains accumulated in the\nwedge\n P9: The number of polymer chains which potentially could diffuse to the wedge-shaped meniscus\nregion\n P9: depended on the number density of polymer chains adsorbed on the substrate surrounding\nthe filament\n P9: suggests the existence of a threshold number of polymer chains which can be included in the\nmeniscus region.\n P9: filaments containing stretched polymer chains were co-deposited together with non-stretched\npolymer chains.\n P10: formation of “kebabs” is only possible if a sufficient number of polymer chains is available\nat its growth site.\n P10: We note that an analogy but also some differences existto observations of substrate-induced\nprefreezing of polymer chains\n P10: we observe stable coexistence of crystalline and molten polymer chains\n P11: we examine the origin of the coexistence of molten and crystalline polymer chains in such\nnanoscopic filaments\n P11: bead-like structures (“kebabs”) were formed only when sufficient molten polymer chains\nwere available.\n P12: In order to form a stable (critical) secondary nucleus, a minimum volume of molten polymer\nchains needs to be available at the growth front of the crystal\n32\n P12: in finite systems like ultra-thin nanometer-sized filaments the number of available molten\npolymer chains is easily exhausted\n P12: Thus, once the number of molten polymer chains in the surrounding\n P12: if the residual number of molten polymer chains at the growth front\n P12: these polymer chains cannot crystallize and will stay in the molten state.\n P17: even after the complete melting of the nanoscopic filaments, the stretched polymer chains\ninside the filament kept some memory which decayed only slowly\nIn the following cases, the word \"polymers\" has been replaced by \"polymer\":\n P4: the crystalline core was covered by a layer of molten polymer which was separated from the\ncore by a well-defined interface\n P4: we analyzed the mechanical properties of filaments and their surrounding either crystalline\nor molten polymer via the phase signal of intermittent-contact mode atomic force microscopy\n(AFM)\n P8: within each PE filament a solid crystalline core coexisted with molten polymer, forming soft\nzones at the boundary of the filament.\n P9: the pressure 𝑝 inside this wedge of molten polymer was lower than the surrounding\nwedge\natmospheric pressure 𝑝 ,\natm\n P9: The latter formed an adsorbed ultrathin layer of liquid-like polymer in the surrounding of the\nfilaments.\n P12: we anticipate that filaments with a crystalline core (blue area in Figure 3a) are surrounded\nwith a narrow zone of molten polymer at the growth front\n P12: we arrive at two predictions for the equilibrium width 𝑊 of the soft zone of molten\nsoft\npolymer\n P12: we assume that a cylindrical secondary nucleus can only form in the soft zone of molten\npolymer at the periphery of a hard core\n P13: we may assume that the geometry of the soft meniscus region containing molten polymer\nis the same for all filaments\n33\n P15: we can conclude that all filaments, including filaments decorated with bead-like structures,\nwere bounded by coexisting soft zones of molten polymer of a size just below a size proportional\nto 𝑟∗(Δ𝑇) of the critical secondary nucleus.\n P16: Besides the limiting size derived from the minimum volume of molten polymer required for\nthe formation of a stable (critical) secondary nucleus at the crystal growth front\n P17: distinguishing even small differences in elastic properties of polymer.\n P17: we conclude that the soft region contained molten polymer which bounded the solid core\nregion\n P17: the solid core region of the filament consisting of the crystallized polymer.\n P18: If the size of the soft zone of molten polymer, characterized by its width 𝑊 (𝑇),\nsoft\nThe text contains various other flaws; here are some examples:\nWe are deeply grateful to the reviewer for the meticulous reading of our manuscript and for\nproviding such a comprehensive list of specific corrections. We agree with all the points raised\nand have carefully revised the text accordingly to address each of the flaws highlighted, including\nthe rephrasing of awkward constructions, clarification of ambiguous statements, correction of\ngrammatical errors and typos, and consistent use of precise terminology.\n P. 1, first line of the Abstract: “is causing that crystals are often” Please rephrase. E.g. “results\nin crystals often being”\nWe have changed that.\n P. 2 below center: “several theoretical approaches” could the authors name some of them here?\nThere were references behind the dot of this sentence as examples.\n(11) Lipowsky, R.; Speth, W. Semi-Infinite Systems with First-Order Bulk Transitions. Phys.\nRev. B 1983, 28 (7), 3983–3993. https://doi.org/10.1103/PhysRevB.28.3983.\n(12) Conde, M. M.; Vega, C.; Patrykiejew, A. The Thickness of a Liquid Layer on the Free\nSurface of Ice as Obtained from Computer Simulation. The Journal of Chemical Physics 2008,\n129 (1), 014702. https://doi.org/10.1063/1.2940195.\n P. 2 last paragraph, “for a finite number N of molecules and a finite volume V”: I believe the\nauthors mean “for a fixed number N of molecules and a fixed volume V”\nWe have changed that.\n34\n P. 3 near the top, “and T are constant”: It would be better to add “while the particle number N\ncan vary/fluctuate” for clarity.\nThe sentence has been changed to “where the chemical potential 𝜇, 𝑉, and 𝑇 are constant while\nthe number 𝑁 of particles in the microstates can vary, is unstable and has been identified as the\ncritical nucleus.”\n P. 3 second paragraph: “In the here presented study” needs to be changed to “In the study\npresented here”. Similarly, for “here applied conditions” on p. 7, “The here investigated\nfilaments” on p. 8, and “here investigated filaments” on p. 10.\nWe have changed all these points.\n P. 3 in the 2nd paragraph, “polymers in ultrathin layers … crystallized very slowly.” I think this\nshould be changed to present tense: “polymers in ultrathin layers … crystallize very slowly.”\nWe have changed that.\n P. 5, third sentence starting with “Naively”. It should be made clear that the “uniform\nmechanical properties” should be considered across rather than along the filaments.\nWe have included this useful information in the following sentence, which helps clarify that the\ndifference in mechanical properties is perpendicular to the filament.\nThe sentence “However, the phase signal derived from the intermittent-contact mode AFM\nmeasurements clearly revealed the coexistence of two regions within the filaments, characterized\nby distinctly different properties.” has been changed to “However, the phase signal derived from\nthe intermittent-contact mode AFM measurements clearly revealed the coexistence of two regions\nacross the filaments, characterized by distinctly different mechanical properties.”\n P. 5, captions of Figure 5: “or a PE-B”.\nThe sentence “The PE-M sample was used here.” has been added to the figure caption of Figure 5.\n P. 5, 2nd line from the bottom can be made clearer: “For an easier comparison of a) with b)\nand c) with d),”\nWe have changed that.\n P. 6: “in the Methods” section.”\nWe have changed that.\n P. 8 right above Figure 3: “coexisted with molten EDGES, soft zones at the LATERAL\nboundaries of the filament.”\nzones at the boundary of the filament.” has been changed to “PE filament a solid crystalline core\ncoexisted with meniscus edges of molten polymer, i.e., soft zones at the lateral boundary of the\nfilament.”\n P. 9: The tense of verbs in the first paragraph should be changed to present tense.\nThe verb tense in that paragraph has been revised to the present tense for sentences that describe\ngeneral facts or established knowledge.\n P. 9, beginning of second paragraph: “SOME with a uniform width and OTHERS decorated”.\nWe have changed that.\n P. 13 top: What is the basis of “the assumption of Wsoft ~ r*”?\n“Moreover, since secondary nuclei adopt the same orientation as the crystal face on which they\nare formed — that is, polymer chains are oriented and aligned along the axis of the crystalline\nnanofilament — a secondary cylindrical nucleus is also aligned along the axis of the nanofilament.\nThere, the radius is in the direction perpendicular to the axis of the filament. Accordingly, we\nassume a relation between the critical radius for nucleation ( 𝑟∗) with the value of 𝑊 , i.e.,\nsoft\n𝑊 ~ 𝑟∗.” has been added to P12 of the manuscript before the sentence “Consequently, we\nsoft\nanticipate that filaments with a crystalline core (blue area in Figure 3a) are surrounded with a\nnarrow zone of molten polymer at the growth front (red area in Figure 3a).”\n P. 13, 4th line: “polymer is not expected to influence neither r* nor Wsoft.” contains too many\nnegatives. Correct: “polymer is expected to influence neither r* nor Wsoft.” or “polymer is not\nexpected to influence either r* or Wsoft.”\nWe have changed that.\n P. 13, below eq.(1): “∆𝑮 is the free energy difference involved in the formation of a crystal\n𝐕\nnucleus”. Dimensional analysis indicates that “∆𝑮 is the free energy change, per volume,” of\n𝐕\nthe formation of a crystal nucleus”. We have changed that.\n P. 13 below eq.(2): “𝝆 is the number density of molecules” is incompatible with later “𝝆 is\n𝐧 𝐧\nranging from ca. 780 kg/m3 to ca. 970 kg/m3”, which shows that “𝝆 is the mass density of the\n𝐧\nnucleus”. P. 13 below eq.(2): “∆𝑯 is the heat of fusion per volume element” is incompatible\n𝐟\nwith “∆𝑯 = 4.1 +- 0.2 kJ/mol of CH2”. Dimensional analysis indicates that ): “∆𝑯 is the heat\n𝐟 𝐟\nof fusion per mass” or “∆𝑯 is the specific heat of fusion”.\n𝐟\n𝑛 𝑓\nof polymer at 𝑇∞ = 141 °C.” has been changed to “𝜌 is the mass density of the polymer, ∆𝐻 is\n𝑚 n f\nthe heat of fusion per unit mass of the polymer at 𝑇∞ = 141 °C. ”\nm\n P. 13: I would propose to write “nm” at the end of the equation instead of the awkward “in\n[nm]”.\nEquation (3) has been changed to:\n(𝑇∞)2\n𝑟∗(Δ𝑇) ≈ 𝑃∙ m (3)\n𝑇∙∆𝑇\nwhere the prefactor 𝑃 =\n2𝜎s\n= (0.1±0.02) nm.\n𝜌m∙∆𝐻\nf\n∙\nFor calculating 𝑃, we used the 𝜎 is ranging from 12 mJ/m2 to 15 mJ/m2; ∆𝐻 ≈ 4.1±\ns f\n0.2 kJ/mol of CH , and 𝜌 is ranging from ca. 780 kg/m3 to ca. 970 kg/m3.\n2 m\n P. 14, “on various filaments, either they had a”: Better: “on various filaments, with either a”\nWe have changed that.\n P. 15, Figure 5: In part c), it would be good to mark the absolute T values on the top axis, to\ncounter the impression that W decreases with increasing temperature.\nWe have changed that.\n P. 15, Figure 5: The font sizes are too disparate. They should be made more uniform (mostly by\nincreasing the size of the small lettering).\nWe have changed that.\n P. 15, caption of Figure 5: The sentence starting with “The two AFM phase images” seems to\nlack a main verb.\nThe two AFM phase images were measured at 𝑇 ≈100 °C, …\n P. 16, right below the table: “and possible intercepts.” should probably be rephrased as “and\nadditive constants.” We have changed that.\n P. 17: “Spin coating experiments were performed” We have changed that.\nThis paper presents an interesting study on the coexistence of a crystal- melt interface on\nsupported polyethylene filaments. I found that the paper is not easy to read, and at times it is\nconfusing. I urge the authors to revise it and make it more understandable.\nWe sincerely thank Reviewer #3 for the invested time and the careful evaluation of our manuscript.\nWe appreciate the positive assessment of our study judging it as interesting and acknowledge the\nconstructive feedback regarding clarity. We agree that the manuscript, in its previous form, was\nnot easy to read and might have been confusing at times. We apologize for these shortcomings and\nundertook a comprehensive revision to significantly improve the organization, flow, and overall\nreadability of the text. We now present our findings in a clearer, more logical, and accessible\nmanner to ensure the message is effectively communicated to the reader.\nThe first phrase of the abstract is quite cryptic and not well written: \"The broken symmetry of\nmolecular interactions at interfaces is causing that crystals are often covered by a thin liquid\nlayer of their own melt.\" No explanation is given to what they refer as broken symmetry and\nthey cannot use the verb in this tense \"is causing\", as this denotes that is happening at this\ninstant...Consider revising and rewriting the abstract to indicate clearly what has been done and\nwhat is new and a significant advance with respect to existing literature.\nWe thank the reviewer for this critical comment and for highlighting the lack of clarity in the\nopening sentence of our abstract. We agree that the original phrasing was cryptic and that the verb\ntense was inappropriate. We aimed to refer to the symmetry of the interactions with neighboring\nmolecules which is different for molecules in the bulk and molecules at interfaces. When going\nfrom a crystalline phase to the surrounding phase (liquid or gas), several symmetry elements\nchange, for example, from a three-dimensionally ordered to a disordered state of translational\ninvariance. One may consider an interface or surface as a planar perturbation reflected by changes\nin (number) density, (two-or three-dimensional) order or translational/rotational symmetry in the\nperpendicular direction. For systems governed by short-range forces, such a perturbation decays\nexponentially. Long range interactions typically cause a slower decay of these perturbations.\nInspired by the idea of broken symmetry introduced by Anderson (Anderson PW. 1972 More is\n38\ndifferent. Science 177, 393–396; doi:10.1126/science.177.4047.393) where he described the\nscale-dependent limitations of symmetrical physical laws for explaining common macromolecular\nstructures, various theoretical treatments have been developed to account for the properties of\ncrystal surfaces leading to concepts of surface melting, premelting or quasi-liquid (molten) layers\n(see for example: R. Lipowsky, Critical Surface Phenomena at First-Order Bulk Transitions,\nPhys. Rev. Lett., 49, 1575 (1982), https://doi.org/10.1103/PhysRevLett.49.1575 , P. Nozières,\nSurface melting and crystal shape. J. de Phys., 1989, 50 (18), pp.2541-\n2550.https://doi.org/10.1051/jphys:0198900500180254100 ; U. Tartaglino, T. Zykova-Timan, F.\nErcolessi, E. Tosatti, Melting and nonmelting of solid surfaces and nanosystems, Phys. Rep.\n411 (2005) 291–321, https://doi.org/10.1016/j.physrep.2005.01.004 ) and appropriate experiments\nwere performed (see for example: Joost W. M. Frenken, Peter M. J. Maree, and J. Friso van der\nVeen, Observation of surface-initiated melting, Physical Review B, 34(11), 7506-7516 (1986);\nhttps://doi.org/10.1103/PhysRevB.34.7506 ; J. G. Dash (1989) Surface melting, Contemporary\nPhysics, 30:2, 89-100, https://doi.org/10.1080/00107518908225509 ; Yang Yang, Mark Asta, and\nBrian B. Laird, Solid-Liquid Interfacial Premelting, Phys. Rev. Lett. 110, 096102 – Published\nAsakawa, Ken Nagashima, Shunichi Nakatsubo, and Yoshinori Furukawa, How do Quasi-Liquid\nLayers Emerge from Ice Crystal Surfaces? Cryst. Growth Des. 13, 1761−1766 (2013),\nhttps://doi.org/10.1021/cg400086j ).\nThus, in order to avoid confusion by trying to capture the complexity of the problem in a short\nsentence, we have revised the opening sentence of our abstract. It reads now as: “An interface or\nsurface may be considered as a planar perturbation reflected by changes in molecular properties in\nthe direction perpendicular to the interface ore surface. As a consequence, predicted by theory and\nshown by experiments, crystals are often covered by a thin liquid layer of their own melt.”\nFurthermore, we undertook a comprehensive revision of the entire abstract to ensure it clearly\nstates the objectives, findings, and significant advances of our work with respect to the existing\nliterature.\nThey write: For all studied PE filaments and independent of their size, the coexistence of a hard\n(solid) core bounded by a soft (liquid-like) layer near the filament-substrate contact line was\nconfirmed.\nmolecular weight microscopic-sized PE fiber or bundles of fibers to have the same molten surface.\nIt would seem that the phenomena observed in most likely present in nanometer fibers.\nWe thank the reviewer for this important question, which allows us to clarify the scope of our claim.\nIn agreement with theoretical predictions and supported by experiments on various systems, including\nsemi-infinite metal surfaces and ice crystals, we believe that our statement that “a soft, liquid-like\nmeniscus layer coexists with a solid core” essentially applies in general and is not limited by size or\ngeometry. In our specific investigation, we have verified that all studied PE filaments ranging from 10\nnm to 100 nm were bounded by an approximately constant soft (liquid-like) layer near the filament-\nsubstrate contact line, with a comparable width, 𝑊 , for all filaments. Furthermore, the same soft\nsoft\nlayer was observed at the periphery of the much larger, bead-like structures that decorated some of\nthese nanofilaments and reached a size up to the micrometer scale. However, while we believe that all\npolymer crystals are bounded by a nanoscopic liquid layer, we anticipate that several factors like\ngeometry and chain folds do have an impact on (and will lead to changes of) the observed value of\n𝑊 . Therefore, we agree that our conclusions based on the observations of the specific system of PE\nsoft\nnanofilaments cannot be generalized without precautions and deviations in the value of 𝑊 have to\nsoft\nbe expected. In particular, for different geometries like lamellar crystals of highly folded crystalline\nstates of polymers we need to perform further dedicated investigations.\nThe sentence \"We notice that, for different geometries like lamellar crystals of highly folded crystalline\nstates of polymers, further dedicated investigations need to be performed.\" has been added at the end\nof the first paragraph after Figure 1.\nIn many instances, the authors use the word \"polymer\" to refer to polymer chains, for instance,\njust one example:\n\"However, in finite systems like ultra-thin nanometer-sized filaments the number of available\nmolten polymers is easily exhausted. Thus, once the number of molten polymers in the\nsurrounding and at the growth front is below this minimum value, we anticipate that crystal\ngrowth will stop.\" There is no way they can use the expression \"number of available molten\npolymers is easily exhausted.\" I imagine they are referring to the available molten chains;\notherwise, I do not understand their meaning. This happens several times in the paper.\n\"polymers.\" We agree that using a more specific terminology such as \"polymer chains\" enhances\nthe clarity and precision of our descriptions. We have carefully revised the manuscript accordingly\nand replaced, whenever appropriate, the general term \"polymers\" with the more accurate \"polymer\nchains\" throughout the text. We hope that we have removed all unintended ambiguity.\n P(age)3: To generate separate polymer crystals of small size, we required a system with a highly\nlimited number of polymer chains.\n P3: Previous experiments23 have shown that polymer chains in ultra-thin layers,\n P3: due to the limited number of polymer chains available in such ultra-thin films,\n P3: the homogeneous nucleation step may be circumvented by introducing filaments consisting\nof stretched and uniquely oriented polymer chains\n P3: a shear-induced change in polymer chain conformation can occur\n P3: i.e., polymer chains undergo a transition from randomly coiled to stretched chains.\n P3: attractive interactions between stretched polymer chains,\n P3: Due to the massive loss of conformational entropy and the high degree of order of the polymer\nchains within the filaments\n P4: polymer chains may crystallize easily within the filament\n P4: Thus, not all polymer chains can undergo such a coil-stretch transition\n P4: filaments of stretched and crystalline chains were surrounded initially by non-stretched\nmolten polymer chains.\n P4: which requires that the number of available polymer chains is above a threshold value.\n P4: we anticipate that crystal growth will stop once the number of remaining molten polymer\nchains is below this threshold value.\n P4: allowing for sufficient mobility of non-crystallized polymer chains but also for crystal growth\n P4: Thus, all polymer chains had a chance to crystallize\n P5: If these filaments would have contained molten polymer chains only\n P9: Thus, as a consequence of the acting capillary forces, polymer chains accumulated in the\nwedge\n P9: The number of polymer chains which potentially could diffuse to the wedge-shaped meniscus\nregion\n41\n P9: depended on the number density of polymer chains adsorbed on the substrate surrounding\nthe filament\n P9: suggests the existence of a threshold number of polymer chains which can be included in the\nmeniscus region.\n P9: filaments containing stretched polymer chains were co-deposited together with non-stretched\npolymer chains.\n P10: formation of “kebabs” is only possible if a sufficient number of polymer chains is available\nat its growth site.\n P10: We note that an analogy but also some differences existto observations of substrate-induced\nprefreezing of polymer chains\n P10: we observe stable coexistence of crystalline and molten polymer chains\n P11: we examine the origin of the coexistence of molten and crystalline polymer chains in such\nnanoscopic filaments\n P11: bead-like structures (“kebabs”) were formed only when sufficient molten polymer chains\nwere available.\n P12: In order to form a stable (critical) secondary nucleus, a minimum volume of molten polymer\nchains needs to be available at the growth front of the crystal\n P12: in finite systems like ultra-thin nanometer-sized filaments the number of available molten\npolymer chains is easily exhausted\n P12: Thus, once the number of molten polymer chains in the surrounding\n P12: if the residual number of molten polymer chains at the growth front\n P12: these polymer chains cannot crystallize and will stay in the molten state.\n P17: even after the complete melting of the nanoscopic filaments, the stretched polymer chains\ninside the filament kept some memory which decayed only slowly\nIn the following cases, the word \"polymers\" has been replaced by \"polymer\":\n P4: the crystalline core was covered by a layer of molten polymer which was separated from the\ncore by a well-defined interface\n42\n P4: we analyzed the mechanical properties of filaments and their surrounding either crystalline\nor molten polymer via the phase signal of intermittent-contact mode atomic force microscopy\n(AFM)\n P8: within each PE filament a solid crystalline core coexisted with molten polymer, forming soft\nzones at the boundary of the filament.\n P9: the pressure 𝑝 inside this wedge of molten polymer was lower than the surrounding\nwedge\natmospheric pressure 𝑝 ,\natm\n P9: The latter formed an adsorbed ultrathin layer of liquid-like polymer in the surrounding of the\nfilaments.\n P12: we anticipate that filaments with a crystalline core (blue area in Figure 3a) are surrounded\nwith a narrow zone of molten polymer at the growth front\n P12: we arrive at two predictions for the equilibrium width 𝑊 of the soft zone of molten\nsoft\npolymer\n P12: we assume that a cylindrical secondary nucleus can only form in the soft zone of molten\npolymer at the periphery of a hard core\n P13: we may assume that the geometry of the soft meniscus region containing molten polymer\nis the same for all filaments\n P15: we can conclude that all filaments, including filaments decorated with bead-like structures,\nwere bounded by coexisting soft zones of molten polymer of a size just below a size proportional\nto 𝑟∗(Δ𝑇) of the critical secondary nucleus.\n P16: Besides the limiting size derived from the minimum volume of molten polymer required for\nthe formation of a stable (critical) secondary nucleus at the crystal growth front\n P17: distinguishing even small differences in elastic properties of polymer.\n P17: we conclude that the soft region contained molten polymer which bounded the solid core\nregion\n P17: the solid core region of the filament consisting of the crystallized polymer.\n P18: If the size of the soft zone of molten polymer, characterized by its width 𝑊 (𝑇),\nsoft\npaper is in many parts not clear enough, and the writing should be improved.\nWe sincerely thank the reviewer for the positive assessment of our experiments and analyses, and\nfor the constructive feedback regarding the clarity of our manuscript. We acknowledge that, in\norder to enhance readability, the writing of many parts of the text required significant improvement.\nWe have undertaken a thorough revision of the entire manuscript, focusing on improving the\norganization, flow, and clarity of the text, ensuring that our findings are presented in a more logical,\ncoherent, and accessible manner.\nThey have employed only PE filaments, but in principle, they could have used other filaments\nwhere they can easily vary the degree of crystallinity and check its influence (i.e., PET). The\nintermolecular interactions in this case are weak Van Der Waal forces (as only PE was\nemployed). What would have happened if a polymer like PEO is used or PCL, in which the\nintermolecular forces are stronger?\nWe thank the reviewer for this insightful comment and for suggesting the possibility of extending our\nstudy to other polymer systems with varying degrees of crystallinity and stronger intermolecular\ninteractions, such as PET, PEO, or PCL. We agree that investigating these materials will provide\nvaluable comparative insights into the role of polymer chemistry and intermolecular forces on surface\npremelting and crystal-melt coexistence.\nHowever, the goal of our present manuscript was (i) to verify the coexistence of a crystalline polymer\nfilament with a bounding layer of molten polymers, (ii) to demonstrate that the thickness (width 𝑊 )\nsoft\nof this layer is independent of size, molecular weight or (some) geometrical features of the filament\nlike the “beads-on-a-string” morphology, and (iii) to show that 𝑊 is increasing as the temperature\nsoft\nis raised. The comparison of the temperature-dependence of 𝑊 with predictions of various\nsoft\ntheoretical concepts suggests or even indicates that these concepts may be closely related, as we discuss\nin our manuscript.\nOf course, in subsequent works, we aim to systematically apply the same methodology to a broader\nrange of polymers to precisely explore the influences of crystallinity and intermolecular force\nstrength/range on the phenomena described here, but also to explore the impact of different geometries\n44\nlike lamellar crystals of highly folded crystalline states of polymers on values of 𝑊 . Thus, the\nsoft\nreviewer’s suggestion provides excellent guidance for, and is fully in line with, the direction of our\nfuture research.\nIs there any molecular weight dependence of the effects observed?\nWe thank the reviewer for raising this important question regarding the potential molecular weight\ndependence of the observed effects. In our manuscript we addressed this point indirectly, by comparing\nsamples of polyethylene (samples PE-M and PE-B) consisting of chains of different lengths and of\nextremely different dispersity. For these samples, we systematically examined the width of the soft\nlayer (𝑊 ) across polyethylene nanofilaments, as summarized in Figure 4. Our analysis revealed that,\nsoft\nacross all the samples of different molecular weights studied, the values of 𝑊 remained consistently\nsoft\nwithin the bounds of experimental uncertainty. This indicates that, under the conditions of our\nexperiments, the phenomenon of crystal-melt coexistence, as characterized by the value of 𝑊 ,\nsoft\nis independent of molecular weight.\n45"}, {"round": 2, "reviewer_comments": {"reviewer_1": "In their revised manuscript and its supplmentary information, the authors responded nicely to my concerns by giving more\nthroughout discussions and additional figures. Thus, I am now positive for the revised version to be accepted in this journal.", "reviewer_2": "Despite their long responses to the reviews, the authors have actually only made few changes to the manuscript, and I\nconsider it still significantly deficient. I thought that the experimental results were interesting, but Reviewer 1 has cast doubts\non those. I consider the theory to be sloppy and unconvincing. The authors still claim, in the Abstract, that they “show that\nthese three phenomena are related”, while they really don’t (which they even admitted in their responses: “We agree that in\nour manuscript we only indicated/suggested the possibility of unifying various concepts, i.e., the concepts of nucleation,\nsurface premelting, and finite-size melting point depression. Of course, a thorough proof of such unification requires a\nprofound theoretical development, which is beyond the scope of this study.”). In that case it is unacceptable to write “we\nshow that these three phenomena are related”. Just because two phenomena have a similar simple T-dependence doesn’t\nshow that they are related. The theory remains at a disappointingly low level, mostly just citing diverse sources (see Table 1\nand the scant text below it). The title is also still inaccurate -- the authors simply refused to change it to “Temperature\ndependence of liquid edge layers of supported polyethylene filaments”.\nThe authors also have not changed the misleading claim that Table 1 shows “the characteristic length scale, represented as l,\nof a liquid layer on the surface of a crystal, as predicted by three different phenomena”, while at least two of the entries are\nactually the limiting thicknesses of crystals. The response claims “regarding the interpretation of characteristic length scales\nin Figure 5c and Table 1. We agree that the caption may have been misleading in its presentation. We have thus improved\nits clarity.” but there is no pertinent yellow highlighted change indicated in either caption (and the caption of Table 1 does\nremain misleading). This seems deceptive.\nOn p. 2, we still read “thickness l of the wetting layer” while the Abstract described “molten regions of rather uniform width,\nWsoft” - how are l and Wsoft different? The response says unconvincingly that one is the symbol for the experimental,\nthe other for the theoretical thickness – where in science do we introduce different symbols like that? It still seems that generic l\nis introduced to hide the fact that sometimes it refers to a crystal thickness and sometimes to the thickness of a liquid wetting layer.\nIf in the table instead of “l ~” the authors wrote “Wsoft ~” followed by a crystallite thickness, the inconsistency would\nbecome apparent. On p. 13, r* is confusingly added to the mix of symbols, but it is treated as l in Table 1.\nNo answer was even attempted to my question regarding premelting: why is it favorable thermodynamically that at the\namorphous-crystalline interface, part of the crystal is eaten away as temperature is raised, quite analogous to the\nobservations in this manuscript? The long “response” simply goes off on tangents.\nI had pointed out that the Introduction does not provide a motivation for this work. While the authors tell us that they “fully\nagree that providing a wide-ranging motivation at the beginning of the Introduction will significantly improve the manuscript\nby framing our work within a broader scientific context. Thus, we have revised the opening paragraph to articulate the\nfundamental questions and the broader implications of studying crystal-melt coexistence, the critical size required for\n(secondary) nucleation and its relation to surface melting and the melting point reduction in nanoscopic polymer filaments.”\nno motivation indicating the importance of this work beyond the “ivory tower” has been added. In a polymer journal this might\nbe acceptable, but in a general journal like Nature Communications it is not.\nI had asked if an observed very thin crystal (a bright region about 7 nm across) at high temperature is compatible with the\nGibbs-Thompson equation, which tells us that such thin crystals melt away, in particular at high T. The authors\nacknowledged the presence of this thin crystal but did not answer my question. They deflected by referring to the filament\nthickness, but the Gibbs-Thompson equation is about the crystal thickness. This thickness needs to be accounted for\nquantitatively. Otherwise the claim of equilibrium in the manuscript title and elsewhere is in serious doubt.\nIn summary, the manuscript remains deficient. It still requires major revisions before it can be considered for publication.", "reviewer_3": "My comments and queries have been adequately answered. I can recommend acceptance, although I would carefully revise\nthe writing to make it clearer and easier to understand."}, "author_response": "Comment #1 of Reviewer #2: Despite their long responses to the reviews, the authors have\nactually only made few changes to the manuscript, and I consider it still significantly deficient.\nResponse of the authors: We are grateful to the reviewer for giving us the opportunity to re-\nemphasize the RELATION between the three phenomena, nucleation, surface premelting and the\nmelting point depression of finite size crystals. We now explicitly write how the broken symmetry\nof intermolecular interactions at interfaces introduces a “common” length scale through the\ncompetition of interfacial energy (“surface term”) and internal energy (“volume term”). However,\nwe respectfully disagree with the assessment that our “manuscript is still significantly deficient”\nand that we “only made few changes to the manuscript”.\nAs documented in the “Highlighted Changes” of our manuscript and the Supporting Information\nas well in our point-by-point response to the reviewers, we have made extensive revisions. These\ninclude: changes in the Abstract; a strengthened Introduction; clarification of theoretical\nframeworks; addition of new experimental data and analysis in the Supplementary Information\n(e.g., Section 8, Fig. SI-5 with added discussion in Section 2); systematic clarification of\nterminology throughout the text (over 40 instances); and direct responses to all technical and\ntheoretical concerns raised by all reviewers. We, together with reviewer #1 and reviewer #3, trust\nthat our response represented substantial improvements and believe that we adequately answered\nall questions.\nComment #2 of Reviewer #2: I thought that the experimental results were interesting, but\nReviewer 1 has cast doubts on those.\nResponse of the authors: We thank the reviewer for finding our results interesting. We note that\nReviewer #1, in her/his final comment on the revised manuscript, stated: “In their revised\nmanuscript and its supplementary information, the authors responded nicely to my concerns by\ngiving more throughout discussions and additional figures. Thus, I am now positive for the revised\nversion to be accepted in this journal.” This indicates that the initial questions of Reviewer #1\nwere fully resolved through our revisions and additions. Thus, the “doubts” of Reviewer #2 are\nnot substantiated by Reviewer #1.\nResponse of the authors: We note that the reviewer's criticism appears to be focused on\nestablished theoretical concepts (nucleation, surface premelting, and the melting point depression\nof finite size crystals), rather than our application of these concepts, which are cornerstones of the\nscience of crystallization, developed and refined over decades by the scientific community and\nsupported by a vast body of literature.\nIn our manuscript, our goal was not to validate these established theories, but to use them to\ninterpret our experimental observations. We have demonstrated in our manuscript that the classical\nnucleation theory provides a consistent interpretation for the observed crystal-melt coexistence in\nsupported PE nanofilaments. This interpretation is further strengthened by our experimental\nconfirmation of the temperature dependence predicted by classical nucleation theory, as detailed\nin our manuscript (Pages 12-16).\nThe reason for involving the other two phenomena is twofold. First, they share the same\nfundamental physical origin with nucleation: the competition between a surface energy term and\na volume energy term. Second, the characteristic length scales derived from all three theories, i.e.,\nℓ ~ the ratio of \"surface term\"/\"volume term\", are compatible with our experimental data for\n𝑊 (𝑇). By showing that our experimental data can be described by the various temperature\nsoft\ndependences of ℓ(𝑇) derived for these three distinct phenomena, we pointed out a possible\nrelationship. This link to the ratio of \"surface term\"/\"volume term\", therefore, serves as a\ncompelling starting point for future investigations.\nTherefore, we believe a general critique of these fundamental theories is not justified. Regardless\nof the intended focus of the criticism, we maintain that our specific application of these theories to\ninterpret our experimental data is both appropriate and insightful. We and the other two reviewers\ncould not identify anything “sloppy and unconvincing” in these concepts and their theoretical\ndescriptions.\nthree phenomena are related”, while they really don’t (which they even admitted in their responses:\n“We agree that in our manuscript we only indicated/suggested the possibility of unifying various\nconcepts, i.e., the concepts of nucleation, surface premelting, and finite-size melting point\ndepression. Of course, a thorough proof of such unification requires a profound theoretical\ndevelopment, which is beyond the scope of this study.”).\nResponse of the authors: There is no contradiction. As we have explicitly and repeatedly written\nin our manuscript, for example, at the end of the Results section (page 18): “all three phenomena\noriginate from the broken symmetry of intermolecular interactions at interfaces and the\ncorresponding competition between interfacial energy and internal energy.” It is this competition\nof interfacial energy (“surface term”) and internal energy (“volume term”) that RELATES all\nthree phenomena. Over the last 50 years and more, the interplay of a “surface term” and a “volume\nterm” has been identified in all experimental studies and introduced in all theoretical descriptions\nof nucleation, surface premelting and the melting point depression of finite size crystals. As the\nreviewer certainly knows very well, the balance of an energy per surface (the “surface term”) and\nan energy per volume (the “volume term”), expressed by the ratio of these two terms, introduces\na length scale “ℓ”. The ratio of “surface term”/“volume term” is at the basis of the size of the\ncritical nucleus (𝑟∗), reflects the thickness (𝑙) of the liquid layer in surface premelting and\ndetermines for crystals of a finite size (𝜎) their melting temperature, i.e., 𝜎 is the length scale that\ncontrols the amount of melting point depression. All existing theoretical descriptions of these three\nphenomena yield a temperature dependence of this length scale “ℓ”. In our manuscript, we have\ncompared the predictions of the temperature dependence of all three length scales (i.e., 𝑟∗, 𝑙 and\n𝜎) with our experimentally observed values of 𝑊 (𝑇). This comparison demonstrated that,\nsoft\nwithin the uncertainty of the measured values of 𝑊 (𝑇), all three phenomena are compatible\nsoft\nand consistent with our observations. In our opinion, this should not surprise the reviewer because\nall these phenomena are reflecting the competition of interfacial energy (“surface term”) and\ninternal energy (“volume term”).\nWe stated in the Abstract: “Filaments smaller than ca. 2∙𝑊 (𝑇) were completely molten. The\n𝑠𝑜𝑓𝑡\nvalues of 𝑊 (𝑇) compared favorably with theoretically predicted characteristic length scales\n𝑠𝑜𝑓𝑡\n3\nin the context of nucleation, surface premelting and the melting point depression of finite size\ncrystals.” This is an observation of an empirical relationship, not a claim of theoretical unification.\nOur admission that (in page 17 of our previous reply to reviewers) “a thorough proof of such\nunification requires a profound theoretical development, which is beyond the scope of this study”\nprecisely acknowledges that. By focusing on the temperature dependence, we are presenting a\nrelation between experimental observations and several theoretical concepts, but not a theoretical\nproof. A detailed theoretical description would require to account for the various parameters\ncharacterizing the geometry of the object (the shape of the filament on the substrate) and the\nvarious molecular interactions, often summarized in the interplay of interfacial tensions at the three\nphase contact line.\nComment #5 of Reviewer #2: In that case it is unacceptable to write “we show that these three\nphenomena are related”.\nResponse of the authors: As we have explicitly and repeatedly written in our manuscript, for\nexample at the end of the Results section (page 18): “all three phenomena originate from the\nbroken symmetry of intermolecular interactions at interfaces and the corresponding competition\nbetween interfacial energy and internal energy.” It is this competition of interfacial energy\n(“surface term”) and internal energy (“volume term”) that RELATES all three phenomena. The\nbalance of an energy per surface (the “surface term”) and an energy per volume (the “volume\nterm”), expressed by the ratio of these two terms, introduces a length scale “ℓ”. Therefore, we\nbelieve it is entirely acceptable to write that “we show that these three phenomena are related”.\nDemonstrating that one experimental dataset can be described by the functional forms of three\ndifferent theoretical phenomena reflects the relationship between them, specifically as they are\ngoverned by same underlying physics (the acting intermolecular forces), which is represented by\nthe ratio of “surface term”/“volume term”. In our manuscript, we have written:\nIn the Abstract: “Altogether, we show that these three phenomena are related and dominated by\nthe intermolecular forces acting at crystal surfaces.”\nAt the end of the Results section (Page 18): “Although these phenomena were often treated\nindependently, our results highlight their fundamental relationship. All three phenomena originate\n4\nfrom the broken symmetry of intermolecular interactions at surfaces and the corresponding\ncompetition between interfacial energy and internal energy.”\nAt the end of the Discussion section (Page 19): “While the physical concepts underlying these\nthree phenomena are different, they all were capable to describe our results of 𝑊 (𝑇) within\n𝑠𝑜𝑓𝑡\nthe uncertainty of the datapoints, highlighting the fact that all these phenomena originate from the\nsame intermolecular interactions at the surface of crystals.”\nComment #6 of Reviewer #2: Just because two phenomena have a similar simple T-dependence\ndoesn’t show that they are related.\nResponse of the authors: We agree that similar functional forms alone should not be considered\nas a conclusive proof of a relation between different physical (theoretical) concepts. However, as\ndescribed in our answer to comment #4 of Reviewer #2, the three phenomena are governed by the\nsame underlying physical interactions (intermolecular forces), typically represented by the ratio of\n“surface term”/“volume term”.\nComment #7 of Reviewer #2: The theory remains at a disappointingly low level, mostly just\nciting diverse sources (see Table 1 and the scant text below it).\nResponse of the authors: As expressed through the title of the manuscript, i.e., “Temperature\ndependence of crystal – melt coexistence for supported polyethylene filaments”, our central\nobjective is to interpret the temperature dependence of the experimentally observed coexistence of\na crystalline and a liquid phase. To this end, we compared our experimental results with predictions\nof well-established existing theoretical concepts [ref. 6-10 of the manuscript], all incorporating the\ncompetition between interfacial energy and internal energy. The essence of these theories can be\nfound in textbooks. [Book1, Book2] We therefore did not repeat all details of these theories in our\nmanuscript.\n[Book1] Muthukumar, M. Nucleation in Polymer Crystallization. In Advances in Chemical\nPhysics; John Wiley & Sons, Ltd, 2003; pp 1–63. https://doi.org/10.1002/0471484237.ch1.\n5\n[Book2] Reiter, G.; Strobl, G. R. Progress in Understanding of Polymer Crystallization; Springer\nScience & Business Media, 2007.\nComment #8 of Reviewer #2: The title is also still inaccurate -- the authors simply refused to\nchange it to “Temperature dependence of liquid edge layers of supported polyethylene filaments”.\nResponse of the authors: We respectfully declined the suggestion because we believe the original\ntitle is more accurate and conceptually significant. In the manuscript, we have explained why a\ncrystal can coexist (in equilibrium) with a liquid layer on its surface. Relying on published works,\nwe also mentioned how the thickness of this layer changes with temperature, and why this\nthickness is related to the ratio of “surface term”/“volume term”.\nBesides, in our previous reply to reviewers, we have provided additional arguments.\nOn Page 16 of our previous reply to reviewers, we have written: “We sincerely appreciate the\nreviewer’s thoughtful suggestion regarding the title of our manuscript. We agree that clarity and\naccuracy are essential. The proposed alternative title, “Temperature Dependence of Liquid Edge\nLayers of Supported Polyethylene Filaments,” certainly captures our key experimental\nobservation. At the same time, we believe that the original title, “Temperature Dependence of\nCrystal–Melt Coexistence for Supported Polyethylene Filaments,” is better suited because it\ncaptures in addition the essential physical phenomenon underlying the formation of the liquid edge\nlayers—namely, the equilibrium coexistence between crystalline and molten phases characterized\nby a length scale which depends on temperature”.\nComment #9 of Reviewer #2: The authors also have not changed the misleading claim that Table\n1 shows “the characteristic length scale, represented as l, of a liquid layer on the surface of a crystal,\nas predicted by three different phenomena”, while at least two of the entries are actually the\nlimiting thicknesses of crystals.\nResponse of the authors: In our manuscript and in our previous reply to the reviewers, we have\naddressed carefully how the thickness of a molten layer on the surface of the crystal is linked with\nthe size of a critical nucleus and the melting point depression of a crystal of finite size. To clarify\nthis also in the caption of the Table 1, we have now specified in this caption that details are given\nin the main text (see answer to Comment #10 of Reviewer #2). Providing all information in the\n6\ncaption would be inappropriate. Here, we present a summary of the explanations given in the main\ntext:\nIn our manuscript on page 13, we explained the relation of 𝑊 with the size of the critical\nsoft\nnucleus, as predicted by classical nucleation theory (CNT): “Moreover, since the secondary nuclei\nadopt the same orientation as the original crystal—that is, the nanofilament in which the polymer\nchains are aligned along its axis—the cylindrical nucleus is aligned lengthwise along the\nnanofilament, while its radius is oriented perpendicularly; this radius is directly correlated with\nthe value of 𝑊 , i.e., 𝑊 ~ 𝑟∗.”\nsoft soft\nIn our manuscript on page 17, we related 𝑊 with the melting point depression of a crystal of\nsoft\nfinite size: “Interestingly, for a given finite size of a filament, characterized by its width 𝑤, such\nan increase of W (𝑇) allows to define a size-dependent melting temperature of a filament\nsoft\n𝑇 (𝑤) ≤ 𝑇 (𝑤 = ∞) = 𝑇∞, a phenomenon often referred to as size-dependent “melting point\nm m m\ndepression” of nanostructures.7,8 In other words, when 2∙𝑊 (𝑇) ≈ 𝑤, then the whole filament\nsoft\nis molten.”\nIn our manuscript on page 17, we also linked 𝑊 with the coexistence of a liquid layer on a\nsoft\ncrystal: … the coexistence of a liquid layer on a crystal is also theoretically predicted 6,11,13 and\nexperimentally verified 6,11,13 for planar surfaces on bulk samples in the context of surface\npremelting. Assuming a specific distance-dependence of intermolecular interactions, various\nfunctional forms of the temperature variation of the characteristic length scale, represented as 𝑙,\nof a liquid layer on the surface of a crystal have been predicted (see Table 1). However, these\napproaches did not account for the influence of geometry and curvature, making a direct\ncomparison with our results problematic.\nIn addition, in our previous reply to the reviewers, we have written on Page 18: “As described in\na large number of publications and also demonstrated in recent computer simulations on\nnanoscopic systems, in equilibrium, a crystal can be covered by a liquid layer of its melt. As shown,\nfor example, in ref (1) of our manuscript, U. Tartaglino, T. Zykova-Timan, F. Ercolessi, E. Tosatti,\nMelting and nonmelting of solid surfaces and nanosystems, Phys. Rep. 411 (2005) 291–321,\nhttps://doi.org/10.1016/j.physrep.2005.01.004 , the thickness 𝑙(𝑇) of this liquid layer can be\n7\nrelated to the wetting behavior described via the corresponding surface and interfacial tensions.\nFor curved interfaces, one has to include additional terms like, for example, a term characterized\nby the Tolman length (see, e.g., P. Montero de Hijes, Jorge. R. Espinosa, Valentino Bianco,\nEduardo Sanz, and Carlos Vega, Interfacial Free Energy and Tolman Length of Curved Liquid-\nSolid Interfaces from Equilibrium Studies, J. Phys. Chem. C, 124, 8795-8805 (2020);\nhttps://doi.org/10.1021/acs.jpcc.0c00816 ). Represented through an “interaction” free energy\nterm 𝑉(𝑙), which includes the various short- and long-range intermolecular forces at work, the\nsolid–liquid and the liquid–vapor interfaces “feel” their mutual presence. As described through\nequation (2) in this above-mentioned paper by Tartaglino et al., the thickness 𝑙(𝑇) of this layer\ndiverges as the melting temperature is approached. Thus, a crystal of finite size 𝜎 will be molten\ncompletely when the molten layer covers the whole size of the crystal., i.e., 𝑙(𝑇) ≥ 𝜎. Consequently,\nsimilar to what we showed in Fig. 5 of our manuscript, the value of 𝑙(𝑇) decides also the melting\ntemperature of finite size crystals.”\nComment #10 of Reviewer #2: The response claims “regarding the interpretation of characteristic\nlength scales in Figure 5c and Table 1. We agree that the caption may have been misleading in its\npresentation. We have thus improved its clarity.” but there is no pertinent yellow highlighted\nchange indicated in either caption (and the caption of Table 1 does remain misleading). This seems\ndeceptive.\nResponse of the authors: We apologize that, according to Reviewer #2, the caption of Table 1\ndoes remain misleading.\nWe have now explicitly repeated in this caption that the balance of an energy per surface (the\n“surface term”) and an energy per volume (the “volume term”), expressed by the ratio of these two\nterms, introduces a length scale “ℓ”. The ratio of “surface term”/“volume term” is at the basis of\nthe size of the critical nucleus (𝑟∗), reflects the thickness (𝑙) of the liquid layer in surface premelting\nand determines for crystals of a finite size (𝜎) their melting temperature, i.e., 𝜎 is the length scale that\ncontrols the amount of melting point depression. All existing theoretical descriptions of these\nthree phenomena yield a temperature dependence of this length scale “ℓ”.\nand Table 1.\nComment #11 of Reviewer #2: “On p. 2, we still read “thickness l of the wetting layer” while the\nAbstract described “molten regions of rather uniform width, Wsoft” - how are l and Wsoft\ndifferent?”\nThe response says unconvincingly that one is the symbol for the experimental, the other for the\ntheoretical thickness – where in science do we introduce different symbols like that?\nResponse of the authors: To avoid any misunderstanding, here we briefly and clearly point out\nthe difference between ℓ and W . In the context of surface premelting, ℓ is representing the\nsoft\ntheoretical thickness of the molten layer on a flat (semi-infinite) crystal with a smooth surface.\nW is our experimentally measured width of the molten polyethylene layer at the meniscus\nsoft\nregion between the polyethylene crystal nanofilament and the mica substrate. However, caused by\nthe difference in geometry and the additional interactions with the substrate, W ≠ ℓ but only\nsoft\nproportional to ℓ, i.e., W ~ ℓ.\nsoft\nThe first few sentences of the paragraph above Table 1 have been changed to:\nWhen comparing 𝑟∗(Δ𝑇), as given by equation (3), with the experimentally measured values\nof 𝑊 (Δ𝑇), because we have to account for the specific geometry of the wedge-shaped soft zone\nsoft\nof molten polymer and the influence of the mica substrate, 𝑊 (Δ𝑇) ≠ 𝑟∗(Δ𝑇), but only\nsoft\nproportional to 𝑟∗(Δ𝑇), i.e., 𝑊 (Δ𝑇)~𝑟∗(Δ𝑇). Thus, a constant prefactor 𝐶 is required to\nsoft CNT\nrelate 𝑊 (Δ𝑇) with 𝑟∗(Δ𝑇), we obtain 𝑊 (Δ𝑇) = 𝐶 ∙𝑟∗(Δ𝑇). Using 𝐶 = 16 and\nsoft soft CNT CNT\nemploying equation (3) for 𝑟∗(Δ𝑇) yields the red line shown in Figure 5c.\nIn the first paragraph after Table 1, the sentence has been changed to:\nHowever, these approaches did not account for the influence of geometry, curvature, and the\ninfluence of the mica substrate, making a direct comparison with our results problematic.\nsometimes it refers to a crystal thickness and sometimes to the thickness of a liquid wetting layer.\nResponse of the authors: In our responses to several previous comments given above in this reply,\nwe have already explained how we related the crystal size to a thickness of a molten layer.\nComment #13 of Reviewer #2: If in the table instead of “l ~” the authors wrote “Wsoft ~”\nfollowed by a crystallite thickness, the inconsistency would become apparent.\nResponse of the authors: The competition of interfacial energy and internal energy RELATES\nall three phenomena. The balance of an energy per surface (the “surface term”) and an energy per\nvolume (the “volume term”), expressed by the ratio of “surface term”/“volume term”, introduces\na length scale “ℓ”. The ratio of these two energies is at the basis of the size of the critical nucleus\n(𝑟∗), reflects the thickness (𝑙) of the liquid layer in surface premelting and determines for crystals\nof a finite size (𝜎) their melting temperature, i.e., 𝜎 is the length scale that controls the amount of\nmelting point depression. All existing theoretical descriptions of these three phenomena yield a\ntemperature dependence of this length scale “ℓ”. In our manuscript, we have compared the\npredictions of the temperature dependence all three length scales (i.e., 𝑟∗(𝑇), 𝑙(𝑇) and 𝜎(𝑇)) with\nour experimentally observed values of 𝑊 (𝑇). This comparison demonstrated that, within the\nsoft\nuncertainty of the measured values of 𝑊 (𝑇), all three phenomena are compatible with our\nsoft\nobservations, i.e., W ~ ℓ.\nsoft\nThe end of the first paragraph of the introduction part has been changed to:\nIn the past, the size- and temperature-dependence of these phenomena have been well studied,\nshowing that the physics behind these three phenomena originates from the broken symmetry of\nintermolecular interactions at interfaces, typically discussed by the balance of an energy per\nsurface (the “surface term”) and an energy per volume (the “volume term”). The ratio of “surface\nterm”/“volume term” introduces a length scale “ℓ” which reflects the key physical concept\nunderlying all three phenomena.\ntreated as l in Table 1.\nResponse of the authors: Please see our answer to the previous comments.\nComment #15 of Reviewer #2: No answer was even attempted to my question regarding\npremelting: why is it favorable thermodynamically that at the amorphous-crystalline interface, part\nof the crystal is eaten away as temperature is raised, quite analogous to the observations in this\nmanuscript? The long “response” simply goes off on tangents.\nResponse of the authors: In our previous reply, we have indicated that the thermodynamic\nanalysis (why is it favorable thermodynamically that at the amorphous-crystalline interface, part\nof the crystal is eaten away as temperature is raised) is already done by many previous researchers.\nBy citing their papers, we have written (on page 24 of our reply): “As discussed in the above\nsection, polymer chains within the nanofilament are almost exclusively oriented parallel to the\nfilament axis. There is no surface covered with chain folds as it is exhibited in lamellar crystals of\nchain-folded polymers. Consequently, in the case of the investigated filaments, molten polymer\nchains are coexisting on the lateral surface of the crystalline core of the filaments, which may be\nidentified as the \"growth front\" of the “oriented” polyethylene crystal, rather than the molten\nphase on the fold surface of lamellar polymer crystals, as reported by numerous other researchers,\ne.g., also by Tanabe, Strobl, and Fischer, Surface melting in melt-crystallized linear polyethylene,\nPolymer, 27, 1147-1153 (1986); https://doi.org/10.1016/0032-3861(86)90001-7 .” Therefore,\nindeed “part of the crystal is eaten away as temperature is raised”.\nThere, we also pointed out the difference between the phenomenon reported in our manuscript and\nthe complementary observations of melting of the fold surface of lamellar crystals, as described\nfor example, in the paper from Tanabe, Strobl, and Fischer.\nWe have written (on Page 24 of our reply): “We note that all concepts discussed above are based\non thermodynamic parameters which apply to equilibrium situations. None of these concepts\ninclude any kinetic (time-dependent) aspects. Our results on 𝑊 (𝑇) show no dependence on the\nsoft\nage of the sample, do not depend on filament size or morphology, and have been shown do be\nindependent of molecular weight and dispersity of the polyethylene used. The interpretation of the\n11\nresults on surface melting by Tanabe et al. and later versions by M. Muthukumar† or by J.U.\nSommer‡ also rely on equilibrium, i.e., thermodynamic concepts.\n† Muthukumar, M. (2007). Shifting Paradigms in Polymer Crystallization. In: Reiter, G., Strobl,\nG.R. (eds) Progress in Understanding of Polymer Crystallization. Lecture Notes in Physics, vol\n714. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-47307-6_1\n‡ Sommer, JU. (2007). Theoretical Aspects of the Equilibrium State of Chain Crystals. In: Reiter,\nG., Strobl, G.R. (eds) Progress in Understanding of Polymer Crystallization. Lecture Notes in\nPhysics, vol 714. Springer, Berlin, Heidelberg.https://doi.org/10.1007/3-540-47307-6_2”.\nComment #16 of Reviewer #2: I had pointed out that the Introduction does not provide a\nmotivation for this work. While the authors tell us that they “fully agree that providing a wide-\nranging motivation at the beginning of the Introduction will significantly improve the manuscript\nby framing our work within a broader scientific context. Thus, we have revised the opening\nparagraph to articulate the fundamental questions and the broader implications of studying crystal-\nmelt coexistence, the critical size required for (secondary) nucleation and its relation to surface\nmelting and the melting point reduction in nanoscopic polymer filaments.” no motivation\nindicating the importance of this work beyond the “ivory tower” has been added. In a polymer\njournal this might be acceptable, but in a general journal like Nature Communications it is not.\nResponse of the authors: We are grateful to the reviewer for giving us the opportunity to re-\nemphasize the motivation of our comparison of our experimental observations with theoretical\nconcepts nucleation, surface premelting and the melting point depression of finite size crystals.\nOur aim is to indicate there is a RELATION between these three phenomena. in order to avoid\nfurther confusion or misunderstandings, we now explicitly mention how the broken symmetry of\nintermolecular interactions at surfaces introduces a “common” length scale through the\ncompetition of interfacial energy (“surface term”) and internal energy (“volume term”). This\ndescription of the relation between the three phenomena has been added at the end of the first\nparagraph in the Introduction-section.\n7 nm across) at high temperature is compatible with the Gibbs-Thompson equation, which tells us\nthat such thin crystals melt away, in particular at high T. The authors acknowledged the presence\nof this thin crystal but did not answer my question. They deflected by referring to the filament\nthickness, but the Gibbs-Thompson equation is about the crystal thickness. This thickness needs\nto be accounted for quantitatively. Otherwise the claim of equilibrium in the manuscript title and\nelsewhere is in serious doubt.\nResponse of the authors: Given that the interplay of the acting intermolecular forces results in\nsurface premelting (and not in nonmelting), a crystal, which has a thickness of ca. 7 nm at low\ntemperatures, is obligatory covered by a liquid layer of thickness 𝑙 when the temperature is\nincreased towards the melting point. As shown, for example, in ref (1) of our manuscript, U.\nTartaglino, T. Zykova-Timan, F. Ercolessi, E. Tosatti, Melting and nonmelting of solid surfaces\nand nanosystems, Phys. Rep. 411 (2005) 291–321, https://doi.org/10.1016/j.physrep.2005.01.004,\nthe thickness 𝑙(𝑇) of this liquid layer can be related to the wetting behavior described via the\ncorresponding surface and interfacial tensions. Depending on the type of interactions, the liquid\nsurface layer may become detectable first at a temperature of about 3/4 of the bulk melting\ntemperature. Thus, for our filaments of a width \"𝑤\", a crystalline core of W = 7 nm can only\ncryst\nexist if 𝑤 ≅ 2∙W (𝑇)+7 nm. Assuming that 𝑤 does not change with temperature 𝑇, the\nsoft\n“bright region about 7 nm across” will be molten when 2∙𝑊 (𝑇) ≈ 𝑤. This is predicted by\nsoft\nthe Gibbs-Thompson equation which, of course, relates a length scale “ℓ” to the ratio of a “surface\nterm” to a “volume term”. Therefore, to specifically answer the reviewer’s question: Yes, a bright\nregion at high T is compatible as long as “ 2∙𝑊 (𝑇) < 𝑤. In other words, when 2∙𝑊 (𝑇) ≈\nsoft soft\n𝑤, then the whole filament is molten.”\nComment #18 of Reviewer #2: In summary, the manuscript remains deficient. It still requires\nmajor revisions before it can be considered for publication.\nResponse of the authors: We hope that we have removed any “doubts” the Reviewer # 2 might\nhave had and that our manuscript can be considered for publication now.\nReviewer #2 (Remarks to the Author):\nComment #1 of Reviewer #2: I continue to perceive the authors’ verbose responses mostly as\nobfuscation. If they contain valid arguments, then those should be added to the manuscript text.\nResponse of the authors: We tried our best to provide all required and valid arguments in our\nmanuscript text. In addition, many supplementary details and profound in-depth discussions\nconcerning the employed theoretical concepts were published in the cited references and\ntherefore were not repeated in the manuscript text.\nComment #2 of Reviewer #2: For instance, I had asked how the presence of a very thin crystal\n(a bright region about 7 nm across) at high temperature is compatible with the Gibbs-Thompson\nequation, which tells us that such thin crystals melt away, in particular at high T. The authors\nhave given a long response and I cannot tell if they’re obfuscating or giving a deep explanation\nthat says that the Gibbs-Thompson equation for the melting point of a thin crystal does not\napply when the crystal is in a filament. They don’t even refer to the Gibbs-Thompson equation\nat all. The response ends in: “Therefore, to specifically answer the reviewer’s question: Yes, a\nbright region at high T is compatible as long as “2 ∙ 𝑊soft(𝑇) < 𝑤. In other words, when 2 ∙\n𝑊soft(𝑇) ≈ 𝑤, then the whole filament is molten.”\nResponse of the authors: We are sorry that Reviewer #2 is not satisfied with our answer. In\nFigure 5 c, we have referred to the Gibbs-Thomson equation, denoted by the black line. In\naddition, the two AFM images measured at 100 °C shown in the inset demonstrate that\nsufficiently small crystals are completely molten, while somewhat thicker crystals still remain,\nas predicted by the Gibbs-Thomson equation. In full agreement with the Gibbs-Thomson\nequation, we have provided a length-scale criterion for the transition from a crystal remaining\nto a crystal completely molten: When 2∙ 𝑊 (𝑇) ≈ 𝑤, then the whole filament is molten.\nsoft\nComment #3 of Reviewer #2: This deflection to “a bright region” rather than a thin crystal\nmakes me suspicious that they may have been evading my question about the stability of a thin\ncrystal at high temperature.\nResponse of the authors: In our response, we have adopted the term “bright region” from the\nprevious Comment #17 of Reviewer #2, which reads: “I had asked if an observed very thin\ncrystal (a bright region about 7 nm across) at high temperature …”. We believe that we have\nnot evaded the reviewer’s question as we showed two AFM images measured at 100 °C in the\n1\ninset of Figure 5 c, which demonstrate that sufficiently small crystals are completely molten,\nwhile somewhat thicker crystals still remain, as predicted by the Gibbs-Thomson equation.\nComment #4 of Reviewer #2: The authors also wrote in their response:\n“Therefore, we believe a general critique of these fundamental theories is not justified.\nRegardless of the intended focus of the criticism, we maintain that our specific application of\nthese theories to interpret our experimental data is both appropriate and insightful. We and the\nother two reviewers could not identify anything “sloppy and unconvincing” in these concepts\nand their theoretical descriptions.”\nThe first and last sentence is a quite obvious (and upsetting) misrepresentation of my evaluation.\nQuite clearly, my critique is not of the theories, but of their superficial presentation and sloppy\nand unconvincing application in this paper.\nResponse of the authors: The aim of this manuscript is not to discuss the three theoretical\nconcepts, which are well known in physics and detailed in the cited references, but to show that\nthe physics behind all three phenomena originates from the broken symmetry of intermolecular\ninteractions at interfaces, reflected by the ratio of 〈surface term/volume term〉 which\nintroduces a length scale “ℓ”. This has been done in a rigorous way.\nComment #5 of Reviewer #2: My conclusion remains that in this paper, talented\nexperimentalists have recorded state-of-the-art images, and then someone added a superficial\ntheoretical sauce mixing various concepts from the literature without a deeper understanding.\nThe authors apparently lack the theoretical background to properly apply these theories, which\nthey admitted in their earlier responses: “We agree that in our manuscript we only\nindicated/suggested the possibility of unifying various concepts, i.e., the concepts of nucleation,\nsurface premelting, and finite-size melting point depression. Of course, a thorough proof of\n**such unification requires a profound theoretical development, which is beyond the scope of\nthis study.**”\nResponse of the authors: We have stated in our manuscript that we have compared the\nexperimentally determined changes in the width with temperature, 𝑊 (𝑇), with the functional\nsoft\ndependence on temperature (mostly given by a scaling law on the difference between melting\ntemperature and sample temperature) predicted for melting point depression, (secondary)\nnucleation and surface premelting. We did not employ any prefactors predicted in the cited\ntheoretical concepts as these values depend on sample geometry, interfacial curvature, and\n2\nsubstrate properties. For example, different results are obtained for differences in the force-\ndistance dependence of the specifically acting intermolecular forces. To implement these\nspecific factors, a more profound theoretical development is required, which is beyond the\nscope of this study.\nComment #6 of Reviewer #2: This limited theoretical understanding was also revealed by\nbasic mistakes in their initial description of the quantities in their equations, which I pointed\nout and they conceded (which, of course, did not raise the level of the authors’ theoretical\nunderstanding).\nResponse of the authors: We are glad that the reviewer agrees that in the revised manuscript\nwe have provided the correct “description of the quantities in the equations”.\nComment #7 of Reviewer #2: Nevertheless, the authors keep stubbornly defending their\nsuperficial conflation of various theories, including their claim, in the Abstract, that they\n“*show* that the three phenomena are related”, with verbose and obfuscating responses but too\nfew substantive changes in the text. They could easily have changed the wording to say that\ntheir analysis *suggests* that the phenomena are related. The theories even have different T-\ndependencies, see Table 1, and “fit” only because the experimental error margins are large, see\nFigure 5(b). I cannot support publication of such unconvincing work of doubtful quality.\nResponse of the authors: As mentioned above, we have employed the concepts of melting\npoint depression, (secondary) nucleation and surface premelting as the underlying physics of\nall three phenomena originates from the broken symmetry of intermolecular interactions at\ninterfaces, reflected by the ratio of 〈surface term/volume term〉 which introduces a length scale\n“ℓ”. We believe that Reviewer #2 agrees that, in this respect, the underlying physics is the same\nfor all three phenomena. Therefore, and as shown in Figure 5 c, all three phenomena exhibit a\nsimilar functional dependence on temperature. To some extent, as shown for two theoretical\npredictions in the context surface premelting (blue lines in Figure 5c), functional dependences\nmay differ for a specific distance-dependence of intermolecular interactions, e.g., screened\nCoulomb interactions (exponential function for the force distance dependence) or van der Waals\ninteractions (often described by a Lennard-Jones power law potential). All this information had\nalready been implemented in the revised version of our manuscript.\nThe authors, based on intermittent AFM imaging, reported the coexistence of crystalline\nphase and molten phase of PE. The sample preparation and AFM imaging were carefully\nperformed and therefore I can find some value in this manuscript, while I found some critical\nerror in interpreting their AFM results. Thus, I cannot recommend for the manuscript in the\ncurrent form to be published in this journal.\nJudging from the temperature-dependent change in W , the authors caught some realistic\nsoft\nchange of those soft molten regions. However, judging from force-distance curves in Fig. SI-\n4, I found that the situation is not so simple.\nThe above figure was clipped from their Fig. SI-4(a). They wrote that the curves 1 and 2 was\nobtained on the mica substrate, while those are not the shape of curves on pure mica. After\nthe jump-in contact, there is a region with almost no change in force value (solid ellipse). This\ntype of regions always observed when surface (or AFM probe itself) is contaminated. This\ncontamination can be the molten state PE. It also can be adsorbed water molecules on mica.\nOr, something else. Or, the intercalation of solvent molecules happened during spin-coating.\nThe probe surface can also be contaminated. The similar doubt was deduced from the\nwithdrawing curve. The long tail (dashed ellipse) will never appear for the contact between\npure mica and clean AFM tip. If so, what is phase zero in their intermittent AFM? The baseline\nis not any more the real baseline. Furthermore, these trends are also observed event at the\ntop of “hard” filaments. How they interpret it? Every discussion made in this manuscript need\nto be modified. Although the figure is not in the main text and therefore this is just a comment,\nbut the vertical axis is somewhat strange. When I converted the force value into deflection\nvalue for curve 10 (assume that the spring constant is 5 N/m), the deflection difference\nbetween D of 5 nm and 10 nm is more than 5 nm. Thus, the sensitivity calibration was wrongly\nperformed.\nSecond, I love their discussion using Fig. 2. Very careful. However, if looking at Figs. SI-4 and\nSI-7, soft layers are not symmetric especially when the scan direction is perpendicular to the\ndirection of the filament. This is surely due to the tip-shape effect. I know this portion is not\nso easy, but the authors should mention something about the effect. If possible, they may be\nable to deconvolute the tip effect and once again discuss the value of W .\nsoft\nFinally, this comment is not so critical but they should clearly describe the resonant frequency\nof the cantilever and actual vibration frequency. They may better to describe how to measure\nthe resonant frequency. Just very far away from the sample surface like written in the manual\nof AFM or perform it at the vicinity of surface? I am asking this question because they used\nvery small amplitude ratio. The resonant frequency changes a lot due to long-range\ninteraction force or air damping. If they were not so careful, all the measurement is not in “soft tapping”\nregion but rather strong force is exerted on the specimen.\nThis paper provides convincing evidence for ~9-nm thin liquid layers along the edges of mica-\nsupported semicylindrical polyethylene filaments, obtained by AFM, and tries to explain their\ntemperature dependence. This is an intriguing system, which was apparently introduced earlier\n(see ref. 57 from 2015).\nThe title “Temperature dependence of crystal – melt coexistence for supported\npolyethylene filaments” does not describe the content of the paper accurately. (Below I’m\nproposing some experiments that could help fulfill the promise of this more ambitious title.) As it\nstands, “Temperature dependence of liquid edge layers of supported polyethylene filaments”\nwould be an accurate title.\nThe theory presented centrally in this paper requires major revisions. Already in the Abstract, the\nauthors name-drop “nucleation, surface premelting, and the melting-point depression of finite-\nsize”, which are later confirmed to refer to the critical nucleus in classical nucleation theory (a\nlower limit of crystal size), premelting, and Gibbs-Thompson theory (another more stringent\nlower limit of crystal size). While at the end of Abstract and elsewhere, the authors claim to have\nunified these phenomena, the theory remains at a disappointingly low level, mostly just citing\ndiverse sources (see Table 1 and the scant text below it). CNT and Gibbs-Thompson theories for\ncrystal size are not really applicable to the thickness of a liquid layer studied here. Specifically,\nin the central Figure 5c, the authors use the two limiting crystal thicknesses to fit the T-\ndependence of a liquid layer. In the caption of Table 1, they make the misleading claim that the\ntable shows “the characteristic length scale, represented as 𝑙, of a liquid layer on the surface of\na crystal, as predicted by three different phenomena”, while at least two of the entries are\nactually the limiting thicknesses of crystals. On p. 2, we read “thickness l of the wetting layer”\nwhile the Abstract described “molten regions of rather uniform width, W ” - how are l and W\nsoft soft\ndifferent? It seems that generic l is introduced to hide the fact that sometimes it refers to a crystal\nthickness and sometimes to the thickness of a liquid wetting layer. If in the table instead of “l ~”\nthe authors wrote “W ~” followed by a crystallite thickness, the inconsistency would become\nsoft\napparent. On p. 13, r* is confusingly added to the mix of symbols, but it is treated as l in Table 1.\nThe theory on p. 13 contains technical flaws in the definition of the quantities (see details\nbelow), which suggests that the authors were struggling with the concepts. It seems that the\nsimple theory on p. 13 is suitable for primary nucleation, but the authors make it clear that they\nwant to consider secondary nucleation. In secondary nucleation, there should be two\ncharacteristic length scales, along the preexisting surface and perpendicular to it, not just a single\nr*. I see this confirmed in the corresponding author’s cartoon of a secondary critical nucleus in\nreference 50, which is one chain stem thick and five stems wide. The assumption of a cylindrical\nsecondary nucleus (top of p. 12) is unreasonable. And why can’t the existing fibrillar crystal\ngrow without nucleation, as in Regime I of Lauritzen-Hoffman theory?\nWhat the data do seem to show is premelting. It is included in Table 1 but not really\ndiscussed. It needs to be explained that while the crystal is the lowest free-energy form of the\npolymer, surface energy of the air-polymer interface makes insertion of a liquid-like layer\nthermodynamically favorable. It should also be discussed that in bulk polymers, premelting has\nbeen invoked, e.g. by Tanabe, Strobl, and Fischer (https://doi.org/10.1016/0032-3861(86)90001-\n7) in studies of HDPE, with a different meaning: without air, at the amorphous-crystalline\ninterface, eating away part of the crystal as temperature is raised, quite analogous to the\nobservations in this manuscript – why is this favorable thermodynamically?\nmelting point, is covered with a liquid-like layer. Why is there no such liquid layer on top of the\ncrystalline filament in Figure 3?\nOne could envision that entanglements that have been excluded from the growing crystal\nand enriched outside the crystal prevent crystallization of the liquid layer. This is invoked by\nsome experts to explain lamellar crystallites alternating with amorphous layers in semicrystalline\npolymers. Is this entanglement mechanism excluded for the liquid layers observed in this study?\nOn p. 2, at the very beginning of the Introduction, the authors do not present a motivation for\ntheir work. I do find it intrinsically interesting, but it would be better if a motivation or grander\nframework was included here.\nFigure 5c displays intriguing images that raise interesting questions. The image at top shows a\nvery thin crystal (the bright region about 7 nm across) at high temperature – is such a small\nthickness compatible with the Gibbs-Thompson equation, which tells us that thin crystals melt\naway, in particular at high T? Have the authors raised temperature and seen the crystal melt\nsuddenly when its thickness drops below the Gibbs-Thompson stability limit?\nAt the bottom of the figure, we see a completely melted filament. Observing it at\ngradually decreasing temperature when at some point a critical nucleus must form would seem\nlike a worthwhile investigation, but the authors do not seem to consider critical nuclei in this\nrelevant context.\nI’m quite uncomfortable with the way in which the authors use the term “polymers” (which they\ndo a lot, 50 times in total). To me, examples of polymers are PE, iPP, and PET. The authors use\n“polymers” where I would prefer “polymer chains”, “polymer segments”, or “polymer”.\nExamples:\nP. 3, center: “the limited number of polymers available” should probably be changed to “the\nlimited number of polymer chains available”.\nP. 3, 5th line from bottom: “interactions between stretched polymers,” should probably be\nchanged to “interactions between stretched polymer chains,”.\nP. 4, first paragraph: “filaments of crystalline chains were surrounded initially by non-stretched\nmolten polymers.” should probably be changed to “filaments of crystalline chains were\nsurrounded initially by non-stretched molten polymer segments.”\nP. 4, 2nd paragraph: “the number of available polymers is above a threshold value.” Should be\nchanged to “the number of available polymer chains [or segments?] is above a threshold value.”\nP. 4, below center: “sufficient mobility of noncrystallized polymers but also” should be changed\nto “sufficient mobility of noncrystallized polymer but also” or “sufficient mobility of\nnoncrystallized polymer segments but also”\nP. 5, second sentence: “contained molten polymers only”\nEtc. etc., including\nP. 18, line 7/8: “contained molten polymers which bounded the solid core region of the filament\nconsisting of crystallized polymers.” should be changed to “contained molten polymer which\nbounded the solid core region of the filament consisting of crystallized polymer.”\nThe text contains various other flaws; here are some examples:\ncrystals often being”\nP. 2 below center: “several theoretical approaches” could the authors name some of them here?\nP. 2 last paragraph, “for a finite number N of molecules and a finite volume V”: I believe the\nauthors mean “for a fixed number N of molecules and a fixed volume V”\nP. 3 near the top, “and T are constant”: It would be better to add “while the particle number N\ncan vary/fluctuate” for clarity.\nP. 3 second paragraph: “In the here presented study” needs to be changed to “In the study\npresented here”. Similarly for “here applied conditions” on p. 7, “The here investigated\nfilaments” on p. 8, and “here investigated filaments” on p. 10.\nP. 3 in the 2nd paragraph, “polymers in ultrathin layers … crystallized very slowly.” I think this\nshould be changed to present tense: , “polymers in ultrathin layers … crystallize very slowly.”\nP. 5, third sentence starting with “Naively”. It should be made clear that the “uniform mechanical\nproperties” should be considered across rather than along the filaments.\nP. 5, captions of Figure 5: “or a PE-B”.\nP. 5, 2nd line from the bottom can be made clearer: “For an easier comparison of a) with b) and c)\nwith d),”\nP. 6: “in the MethodS” section.”\nP. 8 right above Figure 3: “coexisted with molten EDGES, soft zones at the LATERAL\nboundarieS of the filament.”\nP. 9: The tense of verbs in the first paragraph should be changed to present tense.\nP. 9, beginning of second paragraph: “SOME with a uniform width and OTHERS decorated”.\nP. 13 top: What is the basis of “the assumption of W ~ r*”?\nsoft\nP. 13, 4th line: “polymer is not expected to influence neither r* nor W .” contains too many\nsoft\nnegatives. Correct: “polymer is expected to influence neither r* nor W .” or “polymer is not\nsoft\nexpected to influence either r* or W .”\nsoft\nP. 13, below eq.(1): “G is the free energy difference involved in the formation of a crystal\nV\nnucleus”. Dimensional analysis indicates that “G is the free energy change, per volume, ”\nV\nof the formation of a crystal nucleus”.\nn n\nranging from ca. 780 kg/m3 to ca. 970 kg/m3”, which shows that “ is the mass density of the\nn\nnucleus”.\nP. 13 below eq.(2): “H is the heat of fusion per volume element” is incompatible with “H =\nf f\n4.1 +- 0.2 kJ/mol of CH ”. Dimensional analysis indicates that ): “H is the heat of fusion per\n2 f\nmass” or “H is the specific heat of fusion”.\nf\nP. 13: I would propose to write “nm” at the end of the equation instead of the awkward “in\n[nm]”.\nP. 14, “on various filaments, either they had a”: Better: “on various filaments, with either a”\nP. 15, Figure 5: In part c), it would be good to mark the absolute T values on the top axis, to\ncounter the impression that W decreases with increasing temperature.\nP. 15, Figure 5: The font sizes are too disparate. They should be made more uniform (mostly by\nincreasing the size of the small lettering).\nP. 15, caption of Figure 5: The sentence starting with “The two AFM phase images” seems to\nlack a main verb.\nP. 16, right below the table: “and possible intercepts.” should probably be rephrased as “and\nadditive constants.”\nP. 17: “Spin coating experimentS were performed”\n4"}]

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