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Microscopic order and disjoining pressure in lipid films
_CHCMICAL PHYSI_CS LElTERS
Volume 102. number 4
MICROSCOPIC
ORDER
AND DWOINING
PRESSURE
25 November
1983
IN LIPID FILMS
H. WENDEL * and P.M. BISCH Centro Brasileiro de Pesquisas Fisicas - CBPF/CNPq. Rua Xavier Sr‘garrd. I50,22290-Rio de Janeiro. RJ, Brazil Received 11 July 1983
It is demonstrated how, in a phanc tihn at smdll tihn thickness. the orientational order of ?hc hydrocarbon segments of the molecules present in lipid ftims translates into an increase of the macroscopic normal stress from the fti meniscus (film-forming bulk phase) to the proper film phase.
The notion of “disjoining pressure” was introduced as early as the 1930s by Derjaguin and co-workers [ 1] when they explored mechanical methods for quantifying the solvation of solid surfaces immersed in liquids. Later the concept of disjoining pressure was used to discuss the equilibrium properties of free plane soap filns in contact with their film-forming bulk phase (meniscus). Yet, the controversy as to where to localize the pressure jumps making up the disjoining pressure has been going on ever since [24] _ Recently, we have demonstrated [S] that (i) none of the references [24] accounted for the most general conditions of mechanical equilibrium to which the free soap film is subjected, (ii) in general, pressure jumps arise in the transition zone between meniscus and film as weil as between film and external phase, and (iii) the various pressure modulations are related to specific types of interactions we term “electrostaticlike” or “vander-Waals-like” forces, respectively_ In particular, we demonstrated how, in free soap films, the electrostatic-lie force - or, more exactly,
the inhomogeneity of the electric charge distribution - casuses a drop of the equilibrium normal stress from the film to its meniscus. One purpose of the present communication is to extend ref. [S] to the case of a neutral lipidfilm. i.e * On leave of absence from Physik-Department T30. TU Miinchen, D-8046 Carchink and from Abteilung fiir Biophysik, Universitit Ulm, D-7900 Uhn, West Germany_
0 009-2614/83/0000-0000/$03.00
0 1983 North-Holland
a film bare of electric charges. In the light of recent discussion on the repulsion of interfaces due to hydra-
tion forces [6,7], our contribution gains further importance: we will aIso demonstrate how the exact@ measurable microscopic order profile in lipid films finds dkect expresssion in the macroscopic disjoining pressure without being spoilt by other effects as in aqueous films [7]. In order to familiarize the reader with the characteristics of (black) lipid films, we briefly recall some of the basics of our recent theory [8] on the hydrodynamics of such fiL?ls. Unlike soap films, the interior of a lipid f&n does not display any diffuse electric or polarization interface layers whose overlap would stabilize the ftir against collapsing. However, in our theory we still generated an effect similar to the overlap of these layers in soap films by accounting for the order parameter field n(r) intrinsic to each of the lipid “surfactant” layers. The corresponding order parameter describes the orientation of the hydrocarbon chain segments present in the film bulk with respect to the normal to the flat film surface [9] _ As the film surfaces approach each other we found that the previously independent order parameter fields associated with the two interfaces start coupling to each other so as to raise the order close to the film center_ This causes a repulsion of the adjacent film surface thus stabilizing the film against collapse at a critical film thickness_ 361
\‘01u111c I n2. llulllber
CHEWCAL
-I
In formulating our theory mathematically. we generalized the statistical formulation of the theory of interfaces in multicomponent fluid systems [lo] to the treatment of two such interacting interfaces. It is worth mentioning that physically motivated approximations reduced the formal theory to equations similar to those employed by hlarfeljri rmd l&d2 in their piicnoliienologiccl theory of the repulsion of interfaces due to boundary water [G]. Within the fmmework of our model. the stress field at 3 point r in the film is described by the tensor
IT(r) = -4 [nl, I t Vtl@)Vrl(r) - f [W(r)1 ’ 1I Hcrc. n(r) represents
the “escess” the bulk (meniscus) order nb, and X C(r: s](r)). wi:h C(r: q(r)) being direct correlation function. In the
(1)
order at point r over A [n] = ;kTj d3r r2 the Ornstein-Zernike plane film geometry.
g(r) is given by q(r) = q(O) cosh(/3~) = [n,/cosli(~ p11)] cosli(j3~) ,
(2)
where n(O) denotes the “escess” order in the film center over nr,, 71, is the “pseudo-excess” order prevailing at the film surface [S]. and fi = (&~3(77)/an),,,/.-1 [qb] with ~~(77) being the chemical potential of a homogeneous bulk phase of “composition” 77 . We note t173t. exording to eq. (1). the “steric” stress in the center of the mmisms vanishes. There the hydrocarbon segments belonging to lipids or the sol-
vent are distributed BS in a hydrocarbon
isotropicAly and homogeneously bulk solution_
For the following discussion it is important to be aware that. within the model of cqs. (1) and (2), the steric forces derive from 3 potential. i.e. V-n(r)
= - Vo(r) .
(3)
with o(r) = --+A 171~18’ [7)(r)] 2 _
(9
We account for the long-range part of the van der Waals interaction through the van der \VX.I~Spotential - r') p(r’) d’r’ .
h’(r) = Jw(r
with w(r -r’)
= -X/Ir -r’16 density at r. In mechanical equilibrium. VP(r) = -p(r) = -[p(r) 362
(3 [I I], p(r) being the mass
VW(r) + V *n(r)
VW(r) -I- VQ(r)]
21 Novc777ber1983
PIIYSICS LETTERS
(6)
at any point r of the system and for all directions_ The (isotropic) pressure p(r) in eq. (6) allows for those (short-range) interactions not explicitly accounted for in eqs. (l)-(5). Eq. (6) implies that for two points rz and r5 in the film differing only in their respective z coordinates (the z asis being chosen parallel to the normal to the film)
p(r?)
- II,,
+ ~72Wrz)
.
(7)
or P(r2) - fl==(r,)
+ P2 Iv(r2) =PN2 -
Eq. (7) is obtained
by integrating
09
the z component
of eq. (6) along 3. This equation expresses the fact that in a film with constant mass density p3 in mechanicxl equilibrium. no work is required to transfer an infinitesimal volume element parallel to the film normal provided p(r). n(r) and W(r) change according to their equilibrium profiles_ The constant pNz in eq. (S) represents the total normal stress at any point in the film. Similar srguments for the external phase yield p(rr)
+-PI Wt)=pr
,
(9)
with r1 being an arbitrary point in the aqueous phase. Again, the stress is constant and the constant has been chosen to equal the isotron’c pressure p, far away from the interface_ The stress states on either side of a surface relate to each other via the Lapiace condition [5,&l l] [!P - ;izz II = 0,
(10)
with [l l] denoting the jump across the surface of the quantity in the brackets. Combining eqs. (S)-(10) yields Pl - PN? = (PI - P,) w = -H/67rQ
,
(11)
where it” means that eq. (5) is evaluated at the film snrface,H=n2 [X22(& + h,,(P112 - ~PIP$+I, and h is the film thickness. As in a soap film [S], the lipid film exhibits a pressure increase due to the van der Waals interaction when passing from the external phase to the film interior_ In order to guarantee mechanical equilibrium between the meniscus and the film phase, we integrate the x component of eq. (6) from a point in the menis-
CHEhlICAL
Volume 102, number 4
PHYSICS
25 November 1983
LETTERS
cus to a point r2 in the film phase along a path between and parallel to the (plane) fii surfaces_ This procedure yields P2 = P(Q)
+ Q(Q) + P2 W2)
3
(12)
with p2 being the (isotropic) pressure in the meniscus_ We note that Q(r) vanishes identically in the hydrocarbon bulk phase. Combining eqs. (8) and (12) gives PN, - P2 = A [qb] $fi’ $lcosha($31r)
.
(13)
demonstrating that the normal stress increases from the meniscus to the bulk. Physically, the normal stress modulation correlates with the increase of order along the integration path [S]. In eq. (13) this manifests itself by the growing influence of the surface order with decreasing film thickness h (in the meniscus, h = “‘X”). In ref. [S] we showed how the overlap of the diffuse electric layers of a soap film causes a pressure modulation between meniscus and filnl. We claimed that this concept is not restricted to soap films only but is more general. For this reason. we termed the corresponding interactions “electrostatic-like”. Eq. (1) together with eq. (13) displays an explicit example of such an interaction which does not involve any electric charge. Finally, adding eqs. (11) and (13) yields the total disjoining pressure p1 - pz = -H/67&3
+A [7?b] $3’
7&cosh2(:~Iz)
_lo>
5-
O_
!.Ii.I_ ,:,
Ao
b,H,)
_______.___.---•---
-5_
F&LI _Disjoining pressure (solid line) snd its van der Waals (dashdotted line) and steric (dashed line) components lipid films as a function of film thickness.
in
_
(14) In combination with eqs. (11) and (13), this equation deznonstrates that the total disjoining pressure is composed of pressure juznps both across the external phaselftim interface and across the film/meniscus transition zone. The van der Waals contribution [eq. (1 l)], the “steric” contribution [eq. (13)] and the resulting total disjoining pressure [eq. (14)] are shown in fig_ 1 as a function of the film thickness h. The parameters are 81; H= 3.48 X 1 014 erg, A [qb] #VI = 3.225 X 10 s, N/znrn2, and P = 4.55 nm-l _ We see that for Iz> 100 A, the pressure p1 in the bulk electrolyte phase must be smaller than the pressure p, in the film-forming meniscus. Otherwise, the van der Waals interaction would cause the fti to thin. In order to obtain a film of equilibrium thickness h < 100 A, the external pressure must surpass p2_ The corresponding
excess pressure to be applied originates mainly in the “steric” repulsion mechanism which causes a steep rise of the disjoining pressure with decreasing (equilibrium) fh thickness_ In conclusion, we have shown how lipid films rep resent a unique example for linking an QCCWQf& measureable microscopic order profile in a liquid to a mechanically measurable quantity, the disjoining pressure, this effect not being overlayed by the presence of any diffuse change or polarization layers_ We have also shown how the orientational part of this disjoining pressure arises between the &n-forming solution and the proper film phase. We appreciate fmancial support from the CNP,/ KFA (contract INT 30.3-A-L.). HW also appreciates the award of a Heisenberg Fellowship (We 742/4-l). 363
CHEMICAL
References
[ 1J IS.\‘. Derjaguinand XX [ 21 131
14 ] [S] 161
364
Kusstko~, Acta PhysicocNm. URSS 10 (1939) 25. U.V. Toshev and LB. Iranov. Coil. Polym. Sci. 253 (1975) 558. J.A. de Feijter. J.B Rljnbout and A. Vrij. J. Colloid InterfacT Sci. 6-I (1078) 75S. J .C. Ilriksson and B.V. Toshev. Colloids Surfaces 5 (19SZ) 141. P-11. Bosch and 11. \\endel. J. Colloid Interface Sci. to bc published. S. Xtr~cl~.t Jnd N. Radii-. Chcm. Phj s Letters 32 (1976) 119.
PHYSICS LETTERS
75 November
1983
[7] D. Schiby and E. Ruckenstein, Chem. Phys. Letters 9.5 (1983) 435: E. Ruckenstein and D. Schby. Chem. Phys. Letters 95 (19S3) 439. [S] P.M. Bisch and H. Wendel. submitted for publication; H. Wendel, P.M. Bisch and D. GaBez. Colloid Polym. Sci. 260 (1982) 425. [9] J. Seelig and A. See@. Quart. Ret. Biophys 13 (1980) 19. [lo] P.D. I‘leming, A.J.M. Yang and J.H. Gibbs, J. Chem. Phys. 65 (1976) 7. [ll] B.U. Felderhof. J. Chem. Phys. 29 (1968) 44.
Volume 102. number 4
MICROSCOPIC
ORDER
AND DWOINING
PRESSURE
25 November
1983
IN LIPID FILMS
H. WENDEL * and P.M. BISCH Centro Brasileiro de Pesquisas Fisicas - CBPF/CNPq. Rua Xavier Sr‘garrd. I50,22290-Rio de Janeiro. RJ, Brazil Received 11 July 1983
It is demonstrated how, in a phanc tihn at smdll tihn thickness. the orientational order of ?hc hydrocarbon segments of the molecules present in lipid ftims translates into an increase of the macroscopic normal stress from the fti meniscus (film-forming bulk phase) to the proper film phase.
The notion of “disjoining pressure” was introduced as early as the 1930s by Derjaguin and co-workers [ 1] when they explored mechanical methods for quantifying the solvation of solid surfaces immersed in liquids. Later the concept of disjoining pressure was used to discuss the equilibrium properties of free plane soap filns in contact with their film-forming bulk phase (meniscus). Yet, the controversy as to where to localize the pressure jumps making up the disjoining pressure has been going on ever since [24] _ Recently, we have demonstrated [S] that (i) none of the references [24] accounted for the most general conditions of mechanical equilibrium to which the free soap film is subjected, (ii) in general, pressure jumps arise in the transition zone between meniscus and film as weil as between film and external phase, and (iii) the various pressure modulations are related to specific types of interactions we term “electrostaticlike” or “vander-Waals-like” forces, respectively_ In particular, we demonstrated how, in free soap films, the electrostatic-lie force - or, more exactly,
the inhomogeneity of the electric charge distribution - casuses a drop of the equilibrium normal stress from the film to its meniscus. One purpose of the present communication is to extend ref. [S] to the case of a neutral lipidfilm. i.e * On leave of absence from Physik-Department T30. TU Miinchen, D-8046 Carchink and from Abteilung fiir Biophysik, Universitit Ulm, D-7900 Uhn, West Germany_
0 009-2614/83/0000-0000/$03.00
0 1983 North-Holland
a film bare of electric charges. In the light of recent discussion on the repulsion of interfaces due to hydra-
tion forces [6,7], our contribution gains further importance: we will aIso demonstrate how the exact@ measurable microscopic order profile in lipid films finds dkect expresssion in the macroscopic disjoining pressure without being spoilt by other effects as in aqueous films [7]. In order to familiarize the reader with the characteristics of (black) lipid films, we briefly recall some of the basics of our recent theory [8] on the hydrodynamics of such fiL?ls. Unlike soap films, the interior of a lipid f&n does not display any diffuse electric or polarization interface layers whose overlap would stabilize the ftir against collapsing. However, in our theory we still generated an effect similar to the overlap of these layers in soap films by accounting for the order parameter field n(r) intrinsic to each of the lipid “surfactant” layers. The corresponding order parameter describes the orientation of the hydrocarbon chain segments present in the film bulk with respect to the normal to the flat film surface [9] _ As the film surfaces approach each other we found that the previously independent order parameter fields associated with the two interfaces start coupling to each other so as to raise the order close to the film center_ This causes a repulsion of the adjacent film surface thus stabilizing the film against collapse at a critical film thickness_ 361
\‘01u111c I n2. llulllber
CHEWCAL
-I
In formulating our theory mathematically. we generalized the statistical formulation of the theory of interfaces in multicomponent fluid systems [lo] to the treatment of two such interacting interfaces. It is worth mentioning that physically motivated approximations reduced the formal theory to equations similar to those employed by hlarfeljri rmd l&d2 in their piicnoliienologiccl theory of the repulsion of interfaces due to boundary water [G]. Within the fmmework of our model. the stress field at 3 point r in the film is described by the tensor
IT(r) = -4 [nl, I t Vtl@)Vrl(r) - f [W(r)1 ’ 1I Hcrc. n(r) represents
the “escess” the bulk (meniscus) order nb, and X C(r: s](r)). wi:h C(r: q(r)) being direct correlation function. In the
(1)
order at point r over A [n] = ;kTj d3r r2 the Ornstein-Zernike plane film geometry.
g(r) is given by q(r) = q(O) cosh(/3~) = [n,/cosli(~ p11)] cosli(j3~) ,
(2)
where n(O) denotes the “escess” order in the film center over nr,, 71, is the “pseudo-excess” order prevailing at the film surface [S]. and fi = (&~3(77)/an),,,/.-1 [qb] with ~~(77) being the chemical potential of a homogeneous bulk phase of “composition” 77 . We note t173t. exording to eq. (1). the “steric” stress in the center of the mmisms vanishes. There the hydrocarbon segments belonging to lipids or the sol-
vent are distributed BS in a hydrocarbon
isotropicAly and homogeneously bulk solution_
For the following discussion it is important to be aware that. within the model of cqs. (1) and (2), the steric forces derive from 3 potential. i.e. V-n(r)
= - Vo(r) .
(3)
with o(r) = --+A 171~18’ [7)(r)] 2 _
(9
We account for the long-range part of the van der Waals interaction through the van der \VX.I~Spotential - r') p(r’) d’r’ .
h’(r) = Jw(r
with w(r -r’)
= -X/Ir -r’16 density at r. In mechanical equilibrium. VP(r) = -p(r) = -[p(r) 362
(3 [I I], p(r) being the mass
VW(r) + V *n(r)
VW(r) -I- VQ(r)]
21 Novc777ber1983
PIIYSICS LETTERS
(6)
at any point r of the system and for all directions_ The (isotropic) pressure p(r) in eq. (6) allows for those (short-range) interactions not explicitly accounted for in eqs. (l)-(5). Eq. (6) implies that for two points rz and r5 in the film differing only in their respective z coordinates (the z asis being chosen parallel to the normal to the film)
p(r?)
- II,,
+ ~72Wrz)
.
(7)
or P(r2) - fl==(r,)
+ P2 Iv(r2) =PN2 -
Eq. (7) is obtained
by integrating
09
the z component
of eq. (6) along 3. This equation expresses the fact that in a film with constant mass density p3 in mechanicxl equilibrium. no work is required to transfer an infinitesimal volume element parallel to the film normal provided p(r). n(r) and W(r) change according to their equilibrium profiles_ The constant pNz in eq. (S) represents the total normal stress at any point in the film. Similar srguments for the external phase yield p(rr)
+-PI Wt)=pr
,
(9)
with r1 being an arbitrary point in the aqueous phase. Again, the stress is constant and the constant has been chosen to equal the isotron’c pressure p, far away from the interface_ The stress states on either side of a surface relate to each other via the Lapiace condition [5,&l l] [!P - ;izz II = 0,
(10)
with [l l] denoting the jump across the surface of the quantity in the brackets. Combining eqs. (S)-(10) yields Pl - PN? = (PI - P,) w = -H/67rQ
,
(11)
where it” means that eq. (5) is evaluated at the film snrface,H=n2 [X22(& + h,,(P112 - ~PIP$+I, and h is the film thickness. As in a soap film [S], the lipid film exhibits a pressure increase due to the van der Waals interaction when passing from the external phase to the film interior_ In order to guarantee mechanical equilibrium between the meniscus and the film phase, we integrate the x component of eq. (6) from a point in the menis-
CHEhlICAL
Volume 102, number 4
PHYSICS
25 November 1983
LETTERS
cus to a point r2 in the film phase along a path between and parallel to the (plane) fii surfaces_ This procedure yields P2 = P(Q)
+ Q(Q) + P2 W2)
3
(12)
with p2 being the (isotropic) pressure in the meniscus_ We note that Q(r) vanishes identically in the hydrocarbon bulk phase. Combining eqs. (8) and (12) gives PN, - P2 = A [qb] $fi’ $lcosha($31r)
.
(13)
demonstrating that the normal stress increases from the meniscus to the bulk. Physically, the normal stress modulation correlates with the increase of order along the integration path [S]. In eq. (13) this manifests itself by the growing influence of the surface order with decreasing film thickness h (in the meniscus, h = “‘X”). In ref. [S] we showed how the overlap of the diffuse electric layers of a soap film causes a pressure modulation between meniscus and filnl. We claimed that this concept is not restricted to soap films only but is more general. For this reason. we termed the corresponding interactions “electrostatic-like”. Eq. (1) together with eq. (13) displays an explicit example of such an interaction which does not involve any electric charge. Finally, adding eqs. (11) and (13) yields the total disjoining pressure p1 - pz = -H/67&3
+A [7?b] $3’
7&cosh2(:~Iz)
_lo>
5-
O_
!.Ii.I_ ,:,
Ao
b,H,)
_______.___.---•---
-5_
F&LI _Disjoining pressure (solid line) snd its van der Waals (dashdotted line) and steric (dashed line) components lipid films as a function of film thickness.
in
_
(14) In combination with eqs. (11) and (13), this equation deznonstrates that the total disjoining pressure is composed of pressure juznps both across the external phaselftim interface and across the film/meniscus transition zone. The van der Waals contribution [eq. (1 l)], the “steric” contribution [eq. (13)] and the resulting total disjoining pressure [eq. (14)] are shown in fig_ 1 as a function of the film thickness h. The parameters are 81; H= 3.48 X 1 014 erg, A [qb] #VI = 3.225 X 10 s, N/znrn2, and P = 4.55 nm-l _ We see that for Iz> 100 A, the pressure p1 in the bulk electrolyte phase must be smaller than the pressure p, in the film-forming meniscus. Otherwise, the van der Waals interaction would cause the fti to thin. In order to obtain a film of equilibrium thickness h < 100 A, the external pressure must surpass p2_ The corresponding
excess pressure to be applied originates mainly in the “steric” repulsion mechanism which causes a steep rise of the disjoining pressure with decreasing (equilibrium) fh thickness_ In conclusion, we have shown how lipid films rep resent a unique example for linking an QCCWQf& measureable microscopic order profile in a liquid to a mechanically measurable quantity, the disjoining pressure, this effect not being overlayed by the presence of any diffuse change or polarization layers_ We have also shown how the orientational part of this disjoining pressure arises between the &n-forming solution and the proper film phase. We appreciate fmancial support from the CNP,/ KFA (contract INT 30.3-A-L.). HW also appreciates the award of a Heisenberg Fellowship (We 742/4-l). 363
CHEMICAL
References
[ 1J IS.\‘. Derjaguinand XX [ 21 131
14 ] [S] 161
364
Kusstko~, Acta PhysicocNm. URSS 10 (1939) 25. U.V. Toshev and LB. Iranov. Coil. Polym. Sci. 253 (1975) 558. J.A. de Feijter. J.B Rljnbout and A. Vrij. J. Colloid InterfacT Sci. 6-I (1078) 75S. J .C. Ilriksson and B.V. Toshev. Colloids Surfaces 5 (19SZ) 141. P-11. Bosch and 11. \\endel. J. Colloid Interface Sci. to bc published. S. Xtr~cl~.t Jnd N. Radii-. Chcm. Phj s Letters 32 (1976) 119.
PHYSICS LETTERS
75 November
1983
[7] D. Schiby and E. Ruckenstein, Chem. Phys. Letters 9.5 (1983) 435: E. Ruckenstein and D. Schby. Chem. Phys. Letters 95 (19S3) 439. [S] P.M. Bisch and H. Wendel. submitted for publication; H. Wendel, P.M. Bisch and D. GaBez. Colloid Polym. Sci. 260 (1982) 425. [9] J. Seelig and A. See@. Quart. Ret. Biophys 13 (1980) 19. [lo] P.D. I‘leming, A.J.M. Yang and J.H. Gibbs, J. Chem. Phys. 65 (1976) 7. [ll] B.U. Felderhof. J. Chem. Phys. 29 (1968) 44.