Entropy Contribution to the Line Tension: Insights from Polymer Physics, Water String Theory, and the Three-Phase Tension

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 Γ =∼ 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 Γ in the range of 10−5–10−12 N were reported [6–11]. Very few methods allowing experimental measurement of line tension were ∼ √ developed [9–14]. Marmur estimated a line tension as Γ = 4dm γSAγ cot θY, where dm is the molecular dimension, γSA, γ are surface energies of solid and liquid correspondingly, and θY 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 Γ 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 =∼ Γ/γ = 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

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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 Γ ≈ 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 the so- 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 the us solid/liquid/vapor start from the boundary unobvious appears. assumption The line tension that is the 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 Γ may be split into “interactional” (1) (denoted Γint) and entropy-inspired (denoted Γen) 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-phaseΓ = Γ tension.int + Γen 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 “waterΓint stringis due to theory” the interaction presented of molecules and discussed located at in the Ref tripleerence lines with [22 surrounding–24]. X-ray ones, absorption and Γen is the entropy input into the three-phase tension. To estimate Γen, 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 string donating theory” and presented one accepting,and discussed thus in References promoting [22– 24 formation]. 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 surfaces). The (VSFG) authors technique of pointing Referenceto an enhanced [27] exploited ordering the intensity of the vibrational water molecules sum-frequency surrounding generation (VSFG) the hydrophobic technique pointing groups; in particular,to the an enhanced orientation ordering was of demonstrated the water molecules for water surrounding molecules the hydrophobic at the water/polydimethylsiloxane groups; in particular, interfaces,the thus orientation justifying was demonstratedthe chain-like for model water molecules of the triple at the line water/polydimethylsiloxane, shown in Figure 1. interfaces, thus justifying the chain-like model of the triple line, shown in Figure1.

Figure 1. Three-phase line of the sessile droplet is approximated by the polymer chain with a diameter Figure 1. Three-phase line of the sessile droplet is approximated by the polymer chain with a diameter of the monomer dm, where dm is the diameter of the liquid molecule. of the monomer dm, where dm is the diameter of the liquid molecule.

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The use of polymer-chain model for the approximation of the triple enables immediate estimation of the entropy contribution into the line tension, according to Equation (2) (see AppendixA and Reference [28] for details). ∼ kBT Γen = (2) dm where Γen is the entropy force necessary for stretching the pseudo-polymer chain built of liquid −23 J molecules, coinciding with the entropy input to the line tension, kB = 1.38 × 10 K is the Boltzmann constant, T is the temperature and dm is the diameter of the molecule, representing the characteristic dimension of the “monomer” of the “polymer chain” [28]. Assuming for the water molecule ∼ −10 ∼ −11 dm = 2.75 × 10 m we derive from Equation (2) for water at ambient conditions Γen = 1.5 × 10 N. The calculated value of the entropy-inspired line tension Γen is comparable with the experimentally reported values of line tensions reported in References [6,7,9–11,19]. Thus, the input of entropy factors may be at least not negligible in constituting the value of three-phase tension. The value of Γen given by Equation (2) supplies the upper estimation for the entropic contribution to the line tension, because it is based on the “ideal chain approximation”, where the monomers do not interact [28]. For real chains, the interaction between monomers is not negligible and the estimation of the stretching force is supplied by Equation (3):

k T Γ =∼ B (3) en ξ where ξ > dm is the correlation length [28]. It is noteworthy, that the impact of the entropic term into the line tension is necessarily positive (the sign of the line tension remains debatable [5–7,11]). It is ∼ ∼ −12 also recognized that the value of Γen remains considerable for ξ = 5 ÷ 10 dm (values of Γ = 10 N) have been reported [9]. This means that the entropic input into the line tension may be important even when the interaction between molecules forming the three-phase line is considered. Equation (2) also predicts the decrease of the line tension with growth of the size of a liquid molecule dm. Regrettably, experimental data in the field are scarce, and do not enable clear conclusion on this item. The rough estimation of the three-phase tension, given as Equations (2) and (3), is obvious from the dimensional considerations and it has been explicitly mentioned already in Reference [29]. The new aspect strengthened in the presented paper is identification of Equation (2) with the entropic contribution to the entire line tension seen as a sum of “interaction-” and “entropy”-inspired inputs. As with any other “entropic force”, the value of Γen increases with temperature [21,28,30]. Experimental observations suggest the opposite temperature trend: both surface and line tensions are decreased with temperature [10,21,31]. This means that the “interaction” part of the line tension in the studied systems prevails on the “entropic” one. However, the input of entropy-inspired factors into the line tension may be not negligible, as has been suggested for membranes of biological vesicles [9]. At first glance, it seems that the phenomenon of line tension inherent for three-phase systems (liquid lens and sessile droplets) and twin-phase biological vesicles are quite different. However, closer inspection shows that an impact of the gaseous (vapor) phase on the line tension of liquid lenses and sessile droplets is minor. The proximity of the experimental values of the line tension (Γ =∼ 10−10 ÷ 10−11 N) established for biological, vesicles, liquid lens and sessile drops is noteworthy [5–8]. Thus, it is reasonable to suggest that the polymer-chain model is applicable also to biological membranes built from phospholipid molecules containing two hydrophobic fatty acid "tails", promoting orientation effects [25–27].

3. Conclusions The notion of line (three-phase) tension (which is important for constituting apparent contact angles [32,33], contact angle hysteresis [32–35] and stability of Cassie wetting states [17]) is revisited. It is suggested that the line tension is built from “interaction” and “entropic” contribution, which are usually unseparated under calculation of the three-phase tension. For the estimation of the entropic input into line tension the three-phase line is approximated by a polymer chain [28,29]. This model is Entropy 2018, 20, x FOR PEER REVIEW 4 of 6

Entropyinput into2018, line20, 712 tension the three-phase line is approximated by a polymer chain [28,29]. This model4 of is 6 justified by novel experimental data indicating strong orientation effects for liquid molecules located near hydrophobic moieties [22–27]. Thus, the polymer-chain model may be successful for sessile justifieddroplets by placed novel on experimental hydrophobic data (say indicating, polymer) strong solid orientation substrates. effects If the for polymer liquid molecules-chain model located is nearadopted hydrophobic for the triple moieties (three [22-phase)–27]. Thus, line the the entropic polymer-chain contribution model to may the be line successful tension isfor crud sessileely droplets placed on hydrophobic푘퐵푇 (say, polymer) solid substrates. If the polymer-chain model is adopted estimated as Γ푒푛 ≅ , where 휉 is the correlation length [28]. If the triple line is seen as an ideal for the triple (three-phase)휉 line the entropic contribution to the line tension is crudely estimated 푘퐵푇 polymer∼ chain,kBT the entropic input is estimated as Γ푒푛 ≅ , where 푑푚 is the diameter of the liquid as Γen = ξ , where ξ is the correlation length [28]. If푑 the푚 triple line is seen as an ideal polymer chain,molecule, the entropic representing input the is estimated characteristic as Γ dimension=∼ kBT , where of thed is “monomer” the diameter of of the the “polymer liquid molecule, chain”. en dm m representingSimple estimations the characteristic supply dimension for the of entropic the “monomer” input of into the “polymer the line chain”. tension Simple the estimations value of −11 Γ ≅ 1.5 × 10 푁, which is comparable with the reported experimental∼ data−11 [6,7,9,10]. We conclude supply푒푛 for the entropic input into the line tension the value of Γen = 1.5 × 10 N, which is comparable withthat the the entr reportedopic contribution experimental into data the [ 6 line,7,9, 10 tension]. We is conclude not negligible. that the It entropic appears contribution that the entropic into thecontribution line tension to line is nottension negligible. may be Itessential appears and that even the dominating entropic contribution in biological to systems, line tension such as may giant be essentiallipid vesicles, and even where dominating the decisive in biological role of the systems, entropy such input as giant in the lipid elastic vesicles, properties where of the biological decisive rolemembranes of the entropy was demonstrated input in the both elastic experimentally properties of [36 biological,37] and membranes theoretically was [38]. demonstrated Notice that in both the experimentallycase of biological [36 membranes,37] and theoretically the line tension [38]. is Noticenot three that-phase, in the but case the of liquid/liquid biological membranes phases inspire thes linethe phenomenon. tension is not Evansthree-phase, and Rawicz but the stated liquid/liquid in Reference phases [37 inspires] that “in the recent phenomenon. years, it has Evans become and Rawiczapparent stated that in thin Reference condensed [37] that-fluid “in membranes recent years, behave it has become as 2D apparent generalizations that thin of condensed-fluid linear flexible membranespolymers” and behave demonstrated as 2D generalizations that the elastic of linear properties flexible of polymers” vesicles are and entropy demonstrated-dominated. that the elastic properties of vesicles are entropy-dominated. Funding: This research received no external funding.

Funding:AcknowledgmentsThis research: The received author is no indebted external to funding. anonymous reviewers for the instructive reviewing of the paper. Acknowledgments:Conflicts of InterestThes: The author author is indebteddeclares no to anonymousconflict of interests. reviewers for the instructive reviewing of the paper. Conﬂicts of Interest: The author declares no conﬂict of interests. Appendix A. Origin of the Entropic Elastic Forces Appendix A. Origin of the Entropic Elastic Forces Consider isothermal stretching of the polymer ribbon, shown in Figure A1. The Helmholtz free energyConsider of the ribbon isothermal is F stretching= U − TS ( ofU theis the polymer internal ribbon, energy shown of the in rib Figurebon); A1thus,. The the Helmholtz change in freethe energyHelmholtz of the free ribbon energy is underF = U isothermal− TS (U is stretching the internal equals energy [28,29]: of the ribbon); thus, the change in the Helmholtz free energy under isothermal stretching equals [28,29]: Δ퐹 = −푇Δ푆 (A1) Notice, that Δ푆 < 0 takes place (the entropy∆F = − ofT ∆polymerS chains is decreased under stretching(A1) [28,29]), hence Δ퐹 > 0. When the stretch equals dl the modulus of the entropic elastic force fen is given by EqNotice,uationthat (A2)∆: S < 0 takes place (the entropy of polymer chains is decreased under stretching [28,29]), hence ∆F > 0. When the stretch equals dl the modulus of the entropic elastic force f is given by Δ푆 en Equation (A2): |푓푒푛| = −푇 > 0 (A2) ∆푑푙S | fen| = −T > 0 (A2) It is recognized from Equation (A2) that |푓푒푛|~푇dl is true for entropic elastic forces.

It is recognized from Equation (A2) that | fen| ∼ T is true for entropic elastic forces.

(a)

Figure A1. Cont.

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(b) → Figure A1. The origin of the entropic force 푓⃗푒푛 is exemplified with the stretching of the polymer Figure A1. The origin of the entropic force f en is exempliﬁed with the stretching of the polymer ribbon. Sketchribbon. (Sketcha) depicts (a) depicts the non-stretched the non-stretched polymer polymer ribbon. ribbon. Molecules Molecules of polymer of polymer are non-stretched. are non-stretched. Sketch (Sketchb) depicts (b) the depicts same the ribbon same when ribbon stretched when (thestretched stretch (the is dl ). stretch Molecules is dl). of Molecules polymer are of stretched; polymer theare Δ푆 stretched; the entropy of the ribbon S is correspondingly decreased;∆S |푓푒푛| = −푇 > 0. entropy of the ribbon S is correspondingly decreased; | fen| = −T dl > 0. 푑푙

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