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Colloids and Surfaces A: Physicochem. Eng. Aspects 448 (2014) 175–180

Contents lists available at ScienceDirect

Colloids and Surfaces A: Physicochemical andEngineering Aspects

journa l h om epage: www.elsev ier .com/ locate /co lsur fa

ine tension at curved edge of a molecular solid

natoly I. Rusanov ∗, Elena N. Brodskayaendeleev Center, St. Petersburg State University, 199034 St. Petersburg, Russia

i g h l i g h t s

Local cohesive force is calculated ina nano-slit between coaxial cylinderswith dispersion forces.Distribution of local thermodynamicsurface tension on the slit wall isobtained.Line tension at the slit rim is esti-mated as a function of the cylinderradius.

g r a p h i c a l a b s t r a c t

r t i c l e i n f o

rticle history:eceived 2 January 2014eceived in revised form 2 February 2014ccepted 7 February 2014vailable online 17 February 2014

a b s t r a c t

This is the first direct calculation of line tension at a curved edge of a solid. As an example, we consideran empty slit between two unconstrained coaxial cylinders with dispersion forces at nanoscale. Thecomputational scheme includes the calculation of (a) the local stress tensor field inside the slit, (b) localthermodynamic surface tension as a function of the location on the slit walls, and (c) line tension at thecircular rim as a function of the cylinder radius. The calculations have been made on the basis of the

eywords:ine tensionurface tensiontress tensorispersion forces

Irving–Kirkwood stress tensor of statistical mechanics.© 2014 Elsevier B.V. All rights reserved.

ano-slit

. Introduction

The concept of line tension is rooted in Gibbs’ formulation of thelassical theory of capillarity. Every heterogeneous system includesnterfacial surfaces, which can intersect each other to form inter-acial lines. Similarly as an interfacial surface possesses surfaceension, an interfacial line possesses line tension. Depending onhe physical configuration of a system, there are several types of

ine tension. So, first of all, we have to explain what kind of lineension is under consideration. Following the general classificationf line tension [1] (see also surveys [2,3] for solid surfaces), we

∗ Corresponding author. Tel.: +7 812 554 2877; fax: +7 812 428 69 39.E-mail address: rusanov@ar1047.spb.edu (A.I. Rusanov).

ttp://dx.doi.org/10.1016/j.colsurfa.2014.02.019927-7757/© 2014 Elsevier B.V. All rights reserved.

will investigate thermodynamic line tension � as the work of for-mation of a linear interface per unit line length. Quantity � is aone-dimensional analog of thermodynamic surface tension (des-ignated as � by Gibbs). By definition, � is the work of disjoiningtwo pieces of a solid (divided by a plane) from the intermoleculardistance ı to infinity in a vacuum (the cleavage work):

2� =∞∫ı

f (H)dH, (1)

where H is the width of a slit between the pieces and f(H) is the

cohesive force per unit area. The notion of line tension appears asa result of a change in surface tension when approaching an edgealong the surface at the nano-scale. If � is the surface tension valuefar from the edge and �(x) is a function showing the dependence

176 A.I. Rusanov, E.N. Brodskaya / Colloids and Surfaces A:

r R−η

R−ηR

+Rr R−ηr

Rη

1

2

Fai

oa

�

wim

fe�EwWt

E

wsRiod

tdpwiToe�tosc

∇

scd

ig. 1. Scheme of vectors standing in Eq. (3): vector r points the elementary unitrea where the local stress tensor is sought for; the pair of interacting (through thisnit area) molecules are located at points 1 and 2.

f surface tension on the distance x from the edge, the line tensiont the edge is

≡ 1L

∑∫[�(x) − �] dA, (2)

here L is the edge length and A is the surface area, the integrations carried out over each surface and summation over all surfaces

eeting at the edge.As is seen from Eqs. (1) and (2), the knowledge of cohesive force

(H) is necessary for calculating both surface and line tensions. Atvery location and especially near the edge when � is replaced by(x), we may define f(H, x) as a local value of the normal componentN(x) of the stress tensor at the slit surface. The local stress tensoras introduced in statistical mechanics by Irving and Kirkwood [4].orking with a vacuum in a slit, the full Irving–Kirkwood stress

ensor is reduced to its cohesive part of the form

(r) = 12

∑i,j

∫R × R

R˚′

ij(R)dR

1∫0

�(2)ij

(r − �R, r − �R + R)d�, (3)

here R is the vector connecting two interacting particles witheparation R between them and passing through point r (writing×R symbolizes a direct vector product, which is a tensor); �

′ij(R)

s the derivative of the pair interaction potential ˚ij(R) for particlesf sorts i and j (i.e. the force of interaction between the particles);R is the product of Cartesian components dRx, dRy, and dRz; R is

he modulus of vector R; �(2)ij

(r − �R, r − �R + R) is the two-particleistribution function for the particles simultaneously located at theoints r − �R and r − �R + R on the opposite sides from a unit surfaceith coordinate r (that is controlled by an additional variable �). The

ntegration with respect to R is performed over the whole space.he computational scheme of the Irving–Kirkwood tensor accountsnly for those molecular pairs whose connecting lines intersect thelementary unit area with coordinate r (which splits R into fractions

and 1 − �, Fig. 1). The summation is carried out with respect to par-icle sorts (in a multicomponent system) and will be not needed forur calculations. After multiple use, the Irving–Kirkwood stress ten-or has confirmed its physical significance. In particular, it correctlyorresponds to the mechanical equilibrium condition

· E(r) = 0. (4)

A pair potential is an important element of the Irving–Kirkwoodtress tensor. Among various types of forces and numerous cal-ulations made, most attention was primary paid to attractiveispersion forces [5,6], which remain better studied in modern

Physicochem. Eng. Aspects 448 (2014) 175–180

science. For dispersion forces without retardation, the pair poten-tial of molecular interaction is

˚ij(R) = −AijR−6, (5)

where Aij is the interaction constant and R is a distance betweeninteracting molecules of sorts i and j. In our calculations, we assumeEq. (5) to be valid down to R = ı, which attaches a certain model formto the pair potential ˚ij(R).

We now can formulate a computational scheme for line ten-sion. The first stage is calculation of the local normal stress in anempty nano-slit with the aid of the Irving–Kirkwood stress tensor(3). The second stage is calculation of the local surface tension byusing Eq. (1). The third stage includes the estimation of line tensionaccording to Eq. (2). Concerning the first stage, similar calculationsundertaken in the past typically referred to infinite slits (see, e.g.,Ref. [6]). Only recently the results for semi-infinite slit [7,8] yieldedsome output to the line tension of a rectilinear crystal edge [9,10].This paper presents the first attempt to perform calculations for aslit restricted not only in thickness, but also in all lateral directionsand to estimate the line tension of a curved edge as a function ofthe curvature radius. The object for consideration will be an emptycircular slit between two unconstrained coaxial cylinders of sameradius (but, generally, of different nature). A system of such config-uration and also with the van-der-Waals interaction was recentlyinvestigated by Jaiswal and Beaudoin [11], but only with respect tothe integral interaction as a generalization of Hamaker’s classicalapproach [12]. By contrast, our method yields a more detailed pic-ture of distribution of the stress tensor and surface tension over theslit surfaces, which makes possible calculating line tension at thelateral slit rims. A similar detailed investigation, but with respectto local disjoining pressure, was recently done for the slit betweenan infinite plane and a cylindrical body normally oriented to theplane [13].

2. The local cohesive force

Let us detail the first stage of the above computational scheme.The system configuration is shown in Fig. 2 where a is the slit radiusand H is the slit width. Since only a and H are nano-scaled param-eters, we may set the cylindrical bodies to be infinite in length.We specify the point under consideration (where the stress ten-sor is sought-for) on the lower slit wall as the origin of cylindricalcoordinates �, ϕ, z with the z-axis directed above normally to theslit walls. The positions of two interacting molecules in bodies 1and 2 are given by vectors r1 and r2. Evidently, these vectors aredirected oppositely, but are collinear since the straight line seg-ment of length R connecting the two interacting molecules shouldpass through the point under consideration (Fig. 2). We then canwrite

r1 = (1 − �)R, r2 = −�R, (6)

so that R = r1 − r2.Eq. (3) contains the integration with respect to Rx, Ry, Rz, and �.

We change the variables for �1, ϕ1, z1 (the components of vectorr1) and z2 (the z-component of vector r2). The Jacobian modulus forthis transformation is∣∣∣∣ D(Rx, Ry, Rz.�)

D(�1, ϕ1, z1, z2)

∣∣∣∣ = �1(z1 − z2)

z21

.

It also follows from Eq. (6)

R = r1z1 − z2

z1, R = r1

z1 − z2

z1, (7)

A.I. Rusanov, E.N. Brodskaya / Colloids and Surfaces A: Physicochem. Eng. Aspects 448 (2014) 175–180 177

Fw

srt

�

t

wipo

fiCcg

�

wui

�

ic

Fig. 3. The lower circular wall of the slit: C is the wall center, O is the origin ofcylindrical coordinates (OC = r), a is the wall radius, �a1 and �a2 are the ultimate

Eq. (15) allows us to calculate all components of the stress ten-sor as functions of location r on the slit surface and of the slitdimensions a and H, the dependence on r and a being inexplic-itly expressed through functions �1a and �2a given by Eqs. (11) and(12).

ig. 2. A slit between coaxial cylinders 1 and 2: a is the cylinder radius, H is the slitidth, vectors r1 and r2 point the location of two interacting molecules.

where r1 is the modulus of vector r1. From Eq. (7), we also have

R × RR

= r1 × r1

r1· z1 − z2

z1. (8)

We assume the interacting bodies 1 and 2 to be uniform in den-ity. Then the two-particle distribution function �(2)

12 in (3) may beepresented as the product of the one-particle distribution func-ions �(1)

1 and �(1)2

(2)12 = �(1)

1 �(1)2 , (9)

he latter being given as

�(1)1 (z) = c1 (z1 > H), �(1)

1 (z) = 0 (z1 < 0),

�(1)2 (z) = c2 (z2 < 0), �(1)

2 (z) = 0 (z2 > H),(10)

here c1 and c2 are the constant densities of molecular numbersn bodies 1 and 2, respectively. Finally, we simplify the calculationrocedure itself by integrating only over positive values of z andmitting the coefficient 1/2 in Eq. (3).

It remains to set the integration limits for all variables. The limitor ϕ1 is obvious to be 2�, we only have to specify that angle ϕ1s counted off from the straight line connecting the slit wall center

and the point under calculation O (Fig. 3). The limit �1a (at theircumference of radius a) for variable �1 is dependent on ϕ1 andiven by the expression

1a(ϕ1) = −r cos ϕ1 +√

a2 − r2 sin2 ϕ1, (11)

here r is the distance between the slit wall center and the pointnder calculation (Fig. 3). Although quantity �2 has not been

ncluded into the set of variables chosen, its limit �2a (Fig. 3)√

2a(ϕ1) = r cos ϕ1 + a2 − r2 sin2 ϕ1 (12)

s necessary for determining the integration limit for z2 thathanges within the range −f ≤ z2 ≤ 0 where f ≡ z1�2a/�1.

values for �1 and �2 at a given ϕ1.

Accounting for all the said above, Eq. (3) becomes

E = c1c2

2�∫0

dϕ1

�1a∫0

�1d�1

∞∫H

dz1

0∫−f

dz2r1 × r1

r1

× (z1 − z2)2

z31

˚′12

[r1(z1 − z2)

z1

]. (13)

Eq. (13) is valid for any pair potential. Turning now to dispersionforces according to Eq. (5), we have

˚12

[r1(z1 − z2)

z1

]= − A12z6

1

r61(z1 − z2)6

,

˚′12

[r1(z1 − z2)

z1

]= 6A12z7

1

r71(z1 − z2)7

(14)

to change Eq. (13) to the form

E = 6A12c1c2

2�∫0

dϕ1

�1a∫0

�1d�1

∞∫H

z41dz1

0∫−f

dz2

(z1 − z2)5

r1 × r1

r81

· (15)

In this paper, we confine ourselves with finding the normal com-ponent EN ≡ Ezz, which corresponds to the cohesive force f(H, r) inEq. (2). Proceeding to the normal component of stress tensor E bytaking z2

1 from r1 × r1, we have

1 ces A: Physicochem. Eng. Aspects 448 (2014) 175–180

E

E

w

F

G

eclr

E

r

E

w

g

c

F

G

78 A.I. Rusanov, E.N. Brodskaya / Colloids and Surfa

N(r, a, H) = 6A12c1c2

2�∫0

dϕ1

�1a∫0

�1d�1

∞∫H

z61

(�21 + z2

1)4

dz1

×0∫

−f

dz2

(z1 − z2)5. (16)

The subsequent integration over z2, z1, and �1 yields

N(r, a, H) = � A12c1c2

6H3

⎡⎣1 − 3

16�

2�∫0

dϕ F(ϕ, r, a, H) − 9H3

�

2�∫0

dϕ G(ϕ, r, a, H)

⎤⎦ ,

(17)

here

(ϕ, r, a, H) ≡ 2H4(�2

1a + H2)2

− H4

�21a

(�2

1a + H2) + �H3

2�31a

− H3

�31a

arctanH

�1a, (18)

(ϕ, r, a, H) ≡�1a∫0

�51d�1

(�1 + �2a)4

[H

6(

H2 + �21

)3− H

24�21

(H2 + �2

1

)2

− H

16�41

(H2 + �2

1

) + �

32�51

− 1

16�51

arctanH

�1

]. (19)

The integral in (19) can be taken analytically, but the resultingxpression is too large and cumbersome to be written here. In anyase, G(ϕ, r, a, H) is an explicit and easily calculable function. In theimit of an infinite slit (a → ∞), Eq. (17) yields Hamaker’s classicalesult [12]

N = � A12c1c2

6H3. (20)

Introducing dimensionless variables x ≡ H / a and y ≡ r / a, we canepresent Eq. (17) as

N = � A12c1c2

6H3 [1 − g(x, y)] , (21)

here the function

(x, y) ≡ 316�

2�∫0

dϕ F (ϕ, x, y) + 9x3

�

2�∫0

dϕ G (ϕ, x, y) (22)

haracterizes a correction to the classical result, Eq. (20), and

(ϕ, x, y) ≡ 2x4(�2

1a + x2)2

− x4

�21a

(�2

1a + x2) + �x3

2 �31a

− x3

�31a

arctanx

�1a, (23)

(ϕ, x, y) ≡�1a∫

�51d �1

4

[x( )3

− x( )2

0( �1 + �2a) 6 x2 + �2

1 24 �21 x2 + �2

1

− x

16 �41

(x2 + �2

1

) + �

32 �51

− 1

16 �51

arctanx

�1

], (24)

Fig. 4. The dependence of relative local surface tension � (r, a) /� on r for slit dimen-sions a = 10 (curve 1), 20 (2), 40 (3), 60 (4), 80 (5), and 100 (6).

�1a(ϕ, y) ≡ �1a(ϕ, r, a)/a = −y cos ϕ +√

1 − y2 sin2 ϕ, (25)

�2a(ϕ, y) ≡ �2a(ϕ, r, a)/a = y cos ϕ +√

1 − y2 sin2 ϕ, (26)

3. Surface and line tensions

We now can proceed to the calculation of local surface tension.According to Eq. (20), we have for an infinite slit of width H

f (H) = EN(H) = � A12c1c2

6H3(27)

so that Eq. (1) yields a macroscopic value for surface tension

� = �A12c1c2

24ı2. (28)

As for the local surface tension on the circular slit surface, it canbe defined as

2�(r, a, ı) =∞∫ı

EN(r, a, H)dH. (29)

Using Eqs. (16)–(19) and (28), we can accomplish the integrationover the slit width analytically, which results in

� (r, a) = � [1 − g� (a, r)] , (30)

where

g� (a, r) = 38�

⎡⎣2�∫

0

dϕg1 (�1a) + 3

2�∫0

dϕ

�1a∫0

�d �g2 (�)

(� + �2a)4

⎤⎦ , (31)

g1 (�1a) ≡ 1

�21a + 1

+ 1

�21a

+ 1

�31a

(arctan

1�1a

− �

2

), (32)

g2 (�) = 2 − 7

3(

�2 + 1) + 4

3(

�2 + 1)2

+ 1�

(arctan

1�

− �

2

), (33)

and tilde means the ratio of a corresponding quantity to the inter-molecular distance ı: r ≡ r/ı, a ≡ a/ı, and � ≡ �/ı. Fig. 4 exhibitsthe function � (r, a) /� as a result of numerical integration. It isseen that local surface tension in the central part of a circularslit only slightly differs from its macroscopic value �: the devia-

tion is smaller than 1% at r < a − 10ı and is noticeable only on thenanoscale at a < 20ı. However, essential deviations appear at the slitedge where the function � (r, a) /� dramatically decreases givingevidence of existing a negative line tension.

A.I. Rusanov, E.N. Brodskaya / Colloids and Surfaces A:

�

w(ssastgc

4

mwuotiSmah(dOiittrj(tv

t

Fig. 5. Dependence of line tension on the line radius.

In accordance with Eq. (2), we now can calculate line tension as

(a) = 1a

a∫0

[� (r) − �] rdr = 1a

a∫0

� (r) rdr − �a

2. (34)

Substituting Eq. (30) in Eq. (34) yields

�(a)�ı

= −1a

a∫0

rdrg� (a, r) , (35)

here g� (a, r) is given by Eq. (31). The result of integration of Eq.35) is shown in Fig. 5. It is seen that line tension is negative andtrongly dependent on the edge curvature on the nanoscale. Themaller is the curvature radius, the smaller is line tension in itsbsolute value. In the opposite limit of a large radius, line ten-ion tends to its constant value for a rectilinear edge. Earlier [9],his value was approximately estimated as −0.676, which is inood agreement with Fig. 5. This fact confirms the validity of ouralculations.

. Concluding remarks

Although the Irving–Kirkwood stress tensor was suggestedore than 60 years ago, it has not become wide-spread nowadays,hich can be explained by its complex structure and difficulty inse. On the other side, the Irving–Kirkwood stress tensor is not thenly way of introducing a local stress tensor. Generally, a local stressensor is defined not uniquely, which can lead to ambiguous resultsn calculations. This problem was discussed by Ono and Kondo [14],chofield and Henderson [15], and Ljunggren and Eriksson [16]. Theost known alternative to the Irving–Kirkwood stress tensor was

local tensor named the Buff- [14] or Harasima-tensor [15], which,owever, does not satisfy the mechanical equilibrium condition, Eq.4). A great advantage of the Irving–Kirkwood stress tensor is that itoes satisfy Eq. (4) and, therefore, is mechanically self-consistent.f course, it does not mean that the Irving–Kirkwood stress tensor

s unique; it remains to be a local conditional quantity. It is moremportant that the whole force acting on a surface is independent ofhe choice of a local stress tensor. In other words, every local stressensor, including the Irving–Kirkwood tensor, yields a unique trueesult after integration over the surface. But line tension is obtainedust in this way by integrating local surface tension according Eq.34). So we can say that the conditional character of the local stress

ensor can influence local surface tension but leads to a unique truealue of line tension.

The second point for discussion is related to the tempera-ure dependence of line tension. Similarly to surface tension, line

[[[[

Physicochem. Eng. Aspects 448 (2014) 175–180 179

tension should be a temperature dependent quantity. However, thecalculation of line tension was based on the pair potential whosetemperature dependence looks rather problematic. Does it meanthat the result obtained is valid only at zero temperature? Indeed,a pair potential is typically produced by quantum chemistry at zerotemperature, although accounting for temperature is now also metin the literature. If we use a potential from the first principles, we,of course, work at zero temperature and calculate not only linetension but also line energy since free energy and energy coin-cide at zero temperature. Dealing with a quantum chemistry pairpotential, temperature lowers surface tension through decreasingconcentration (density) and increasing the intermolecular distanceı that is the integration limit in our calculations, but these are onlyslight changes. A much more temperature dependent pair poten-tial is expected from statistical mechanics, where the effective pairpotential is an artificial construction replacing many-body interac-tions and corresponding molecular correlations. Working with theIrving–Kirkwood stress tensor at an arbitrary temperature impliesthat such a corresponding pair potential has been given. Then,considering the process of formation of a new surface at any giventemperature, our computational scheme yields corresponding sur-face and line tensions.

It is of note that the Irving–Kirkwood stress tensor was elabo-rated within the statistical mechanics of fluids and includes a binarydistribution function. The application of the Irving–Kirkwood stresstensor to solids, as we did in this article, turns to be much simplerbecause of no necessity in finding the binary distribution function.We calculated thermodynamic line tension of a solid in a vacuum.Generalization of our calculations for a solid in an arbitrary fluidmedium will look more complicated since it will include the binarydistribution function. As for solids in a vacuum, further calcula-tions can be related to the dependence of the line tension of a solidedge on the edge angle. Another line of development is the vari-ety of molecular interactions. The Irving–Kirkwood stress tensor issuitable for any pair potential, but we considered only dispersionforces. Performing calculations for other types of forces makes achallenge for theory.

As for possible application of the results obtained, it is the sameas for the line tension of solids at all. The most promising area ofapplication lies in the theory of solid strength, which found a newsounding after introducing the line tension of cracks [17–21] (seealso surveys [22,23]). The 2D theory of cracking uses line tension fora rectilinear crack front line, but the 3D theory requires line tensionfor a curved front line.

Acknowledgments

This work was supported by the program “Leading ScientificSchools of Russian Federation” (Grant 2744.2014.3) and the RussianFoundation for Basic Research (Grant 13-03-01081).

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[

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80 A.I. Rusanov, E.N. Brodskaya / Colloids and Surfa

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[[

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