Entropy Contribution to the Line Tension: Insights from Polymer Physics, Water String Theory, and the Three-Phase Tension
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entropy Article Entropy Contribution to the Line Tension: Insights from Polymer Physics, Water String Theory, and the Three-Phase Tension Edward Bormashenko Engineering Faculty, Chemical Engineering, Biotechnology and Materials Department, Ariel University, P.O. BOX 3, Ariel 407000, Israel; [email protected] Received: 3 August 2018; Accepted: 13 September 2018; Published: 16 September 2018 Abstract: The notion of three-phase (line) tension remains one of the most disputable notions in surface science. A very broad range of its values has been reported. Experts even do not agree on the sign of line tension. The polymer-chain-like model of three-phase (triple) line enables rough estimation of entropic input into the value of line tension, estimated as G =∼ kBT =∼ 10−11N, en dm where dm is the diameter of the liquid molecule. The introduction of the polymer-chain-like model of the triple line is justified by the “water string” model of the liquid state, predicting strong orientation effects for liquid molecules located near hydrophobic moieties. The estimated value of the entropic input into the line tension is close to experimental findings, reported by various groups, and seems to be relevant for the understanding of elastic properties of biological membranes. Keywords: line tension; entropic contribution; entropic force; orientation effect; hydrophobic substrate 1. Introduction Surface tension is due to the special energy state of the molecules at a solid or liquid surface [1–4]. Molecules located at the triple (three-phase) line where solid, liquid, and gaseous phases meet are also in an unusual energy state [1–4]. The notion of line (three-phase) tension has been introduced by Gibbs. Gibbs stated: “These (triple) lines might be treated in a manner entirely analogous to that in which we have treated surfaces of discontinuity. We might recognize linear densities of energy, of entropy, and of several substances which occur about the line, also a certain linear tension” [5]. Notice that Gibbs emphasized the role of the entropy density, which will play the main role in our approach to the problem of line tension. Even though the concept of line tension is intuitively clear, it remains one of the most obscure and disputable notions in surface science [6–8]. Researchers disagree not only on the value of line tension, but even on its sign. Experimental values of line tension G in the range of 10−5–10−12 N were reported [6–11]. Very few methods allowing experimental measurement of line tension were ∼ p developed [9–14]. Marmur estimated a line tension as G = 4dm gSAg cot qY, where dm is the molecular dimension, gSA, g are surface energies of solid and liquid correspondingly, and qY is the Young angle. Marmur concluded that the magnitude of the line tension is less than 5 × 10−9 N, and that it is positive for acute and negative for obtuse Young angles [15,16]. However, researchers reported negative values of line tension for hydrophilic surfaces [14]. As for the magnitude of line tension, the values in the range 10−9–10−12 look realistic. Large values of G reported in the literature are most likely due to contaminations of the solid surfaces [3]. Let us estimate the characteristic length scale l at which the effect of line tension becomes important by equating surface and “line” energies: l =∼ G/g = 1 − 100nm. The effects relating to line tension can be important for nano-scaled droplets or for nano-scaled rough surfaces. However, these effects also Entropy 2018, 20, 712; doi:10.3390/e20090712 www.mdpi.com/journal/entropy Entropy 2018, 20, x FOR PEER REVIEW 2 of 6 Let us estimate the characteristic length scale l at which the effect of line tension becomes important by equating surface and “line” energies: l / = 1−100 nm. The effects relating to line tension can be important for nano-scaled droplets or for nano-scaled rough surfaces. However, these effects also may be important for the design of microfluidics circuits [6] and stabilization of the Cassie air-trapping wetting regime, enabling manufacturing of superhydrophobic and superoleophobic surfaces [17]. It was also shown that at millimeter-length scale, the gravitational potential provides a gravitational contribution Γ ≈ 1–10 µN, which is always positive [10]. The notion of line tension remains highly controversial, at least due to the conceptual difficulties, which arise because theEntropy interfaces2018, 20, 712 between two phases are always diffuse and never sharp [18]. In 2the of 6 present article we try to estimate the role of the entropy contribution in constituting line (three-phase) tension. may be important for the design of microfluidics circuits [6] and stabilization of the Cassie air-trapping 2. “Waterwetting String regime, Theory”, enabling Insights manufacturing from Polymer of superhydrophobic Physics and andEntropy superoleophobic Contribution surfaces into [17 ]. Line TensionIt was also shown that at millimeter-length scale, the gravitational potential provides a gravitational contribution G ≈ 1–10 µN, which is always positive [10]. The notion of line tension remains highly It shouldcontroversial, be mentioned at least due that to the calculation conceptual difficulties,of line tension which from arise because first principles the interfaces by between MD and DFT (density functionaltwo phases are theory) always simulations diffuse and never is not sharp a [trivial18]. In the task present [19,20]. article Nosonovsky we try to estimate in his the recent role of article argued thatthe both entropy energy contribution and entropy in constituting contribution line (three-phase) should be tension. taken into account for predicting surface and line tensions2. “Water String [21]. Theory”,However, Insights the entropy from Polymer input Physics remains and Entropy “unseparated” Contribution within into Line the Tension reported MD and DFT calculationsIt should beof mentionedthe three- thatphase calculation tension of[19 line,20]. tension We will from try first to principles perform bya rough MD and estimation DFT of input of the(density entropy functional factors theory) in the simulations entire value is not of a trivialthe line task tension. [19,20]. Nosonovsky We restrict in our his recenttreatment article by case, when a sessileargued thatdrop both sits energy on hydrophobic and entropy contribution (say polymer) should besurface, taken into and account the three for predicting-phase tension surface at the solid/liquid/vaporand line tensions boundary [21]. However, appears. the The entropy line input tension remains is “unseparated”also inherent within for theso- reportedcalled liquid MD lens, arising atand the DFTliquid/liquid/vapor calculations of the three-phaseboundary tension[12,18] [and19,20 ].giant We willbiological try to perform vesicles, a rough where estimation a lipid bilayer of input of the entropy factors in the entire value of the line tension. We restrict our treatment by membranecase, in whenits liquid a sessile state drop has sits the on properties hydrophobic of (say a two polymer)-dimensional surface, and liquid the three-phase [8]. tension at Let theus solid/liquid/vaporstart from the boundaryunobvious appears. assumption The line tension that isthe also line inherent tension for so-called Γ may liquid be lens, split into “interactional”arising at(denoted the liquid/liquid/vapor Γ푖푛푡) and entropy boundary-inspired [12,18] and(denoted giant biological Γ푒푛) contributions: vesicles, where a lipid bilayer membrane in its liquid state has the properties of a two-dimensional liquid [8]. Let us start from the unobvious assumptionΓ = Γ that푖푛푡 + theΓ푒푛 line tension G may be split into “interactional” (1) (denoted Gint) and entropy-inspired (denoted Gen) contributions: where Γ푖푛푡 is due to the interaction of molecules located at the triple line with surrounding ones, and Γ푒푛 is the entropy input into the three-phaseG = Gtension.int + Gen To estimate Γ푒푛 , assume that molecules(1) constituting the triple line form the quasi-polymer chain, as depicted in Figure 1. This idea originates from thewhere “waterGint stringis due totheory” the interaction presented of molecules and discussed located at in the Ref tripleerence lines with [22 surrounding–24]. X-ray ones,absorption and Gen is the entropy input into the three-phase tension. To estimate Gen, assume that molecules spectroscopyconstituting and X the-ray triple Raman line form scattering the quasi-polymer demonstrated chain, as that depicted water in Figureconsists1. This of ideastructures originates with two strong Hfrom-bonds, the “water one stringdonating theory” and presented one accepting, and discussed thus in References promoting [22– 24formation]. X-ray absorption of chain-like structuresspectroscopy [22–24]. There and X-ray is much Raman theoretical scattering and demonstrated experimental that water evidence consists that of structureswater molecules with are strongly orientedtwo strong near H-bonds, hydrophobic one donating moieties and one[25– accepting,27], and this thus is promoting the case in formation our treatment of chain-like (recall, that we restrictedstructures our consideration [22–24]. There is by much sessile theoretical droplets, and experimentalplaced on hydrophobic evidence that watersurfaces). molecules The areauthors of strongly oriented near hydrophobic moieties [25–27], and this is the case in our treatment (recall, that we Referencerestricted [27] exploited our consideration the intensity by sessile vibrational droplets, placed sum on-frequency hydrophobic generation