- Email: [email protected]
Derivation of the refractive index of lipid monolayers at an air-water interface
Optical Materials 93 (2019) 1–5
Contents lists available at ScienceDirect
Optical Materials journal homepage: www.elsevier.com/locate/optmat
Derivation of the refractive index of lipid monolayers at an air-water interface
T
A. Gadomskia, N. Kruszewskaa,∗, J.M. Rubib a b
UTP University of Science and Technology, Institute of Mathematics and Physics, Kaliskiego 7, PL–85796, Bydgoszcz, Poland Universitat de Barcelona, Departament de Fisica de la Materia Condensada, Diagonal 647, 08028, Barcelona, Spain
ARTICLE INFO
ABSTRACT
Keywords: Langmuir thin film Soft optical materials Van der Waals type equation of state Kelvin-Kubo law for dielectric spherical cloud Refractive index Lipids
We demonstrate by means of a simple thought experiment on lipid films dispersed over an air-water interface that a core refractive index of the system approaches 2 , a very significant value for the lipid (DPPC or DPPE) monolayers. The thought experiment consists of balancing two principal types of acting pressures: the vertical and the horizontal/lateral ones, both taken at a thermodynamic equilibrium. The horizontal pressure of (modified) van der Waals type comes from the minimalistic application of the (movable) lateral barrier, as can be expected to occur in Langmuir experiments, provided that the application of pressure is very weak. On the other hand, after relaxing the system when the barrier goes back, one expects to arrive at another scenario. This is when the vertical pressure of the air-water quasi-planar system of hydrophobic propensity applies, provided that an idealized assumption of the equally distributed pressurizing energy is true. This is in accord with a Kubo evaluation for a certain number of charged point-like objects immersed in water spherical shells and surrounded by air for which the Kelvin (J.J. Thomson) law prevails. At the horizontal vs. vertical pressure conditions one may uncover, by employing simple analytic means the basic dielectric and optical properties of the domain-wise, pairwise-interaction involving lipid film. It turns out that a simple Gladstone-Dale scenario, pointing to splitting 1 + 0.41 overwhelms, with the fractional value of ca. 0.41 attributed to a the core refractive index into 2 molecular contribution of the charged lipid-water (dipolar) local system, affecting the overall pressure-addressing scenario. (By the value of refractive index centering at one, the vacuum conditions are to be addressed.) One may argue, however, that the fractional part of the refractive index seems to be a bit outside the contemporary experimental reach. On the other hand, it can be rationalized by the Casimir critical soft-matter fluctuational effect.
1. Introduction Lipid monolayers dispersed over an air-water interface are widely accepted as an interesting model system prone to a plethora of phase changes, resulting in an emergence of microdomains in its compressed lipid phase submerged in water [1–6]. There are many applications of lipid systems with versatile physicochemical characteristics, especially in biosensoring, thin biofilms’ formation, bionanotechnology, pharmacology, biomaterials engineering and biomolecular crystal technology, and the likes [2,7,8]. There is also a plausible extension of these applications toward biophysics and cell biology, and in particular model biomembrane formation, with its kinetics and thermodynamics that are of vital biomedical interest [9–11]. In this short communication, we consider an effectively two-
∗
dimensional system at the air-water interface. The system is an amphiphilic (viz lipid) monolayer thin film of Langmuir type (cf. Fig. 1) [12]. In order to achieve the attraction between the amphiphilic molecules’ heads by getting them closer to each other, a weak (oscillating) pressure, provided by a barrier, is applied. The barrier is equipped with a labile spring with a very small spring constant (suitable for standardtemperature and/or close-to-physiologic conditions). This way, there is a chance for amphiphilic molecules to get (locally) closer to each other, and thanks to Casimir-type effect [13,14], providing critical fluctuations in the system, to eventually create small drops or microdomains. The (compressing) pressure coming from the barrier is balanced by the pressure of the extending type, caused by van der Waals forces. Vectors of these two (compressing and extending) forces have a horizontal direction (cf., Fig. 1). In vertical direction, there are also two competing pressures: the one coming from the air and the second - from
Corresponding author. E-mail address: [email protected] (N. Kruszewska).
https://doi.org/10.1016/j.optmat.2019.04.042 Received 30 October 2018; Received in revised form 15 April 2019; Accepted 20 April 2019 Available online 10 May 2019 0925-3467/ © 2019 Elsevier B.V. All rights reserved.
Optical Materials 93 (2019) 1–5
A. Gadomski, et al.
Fig. 2. Simplified sketch of a drop (charged sphere) of radius r half-submerged in a water. The drop consists of one charged amphiphilic molecule's head surrounded by water molecules. Based on Kelvin's equation there is a difference between pressures inside ( pw ) and outside ( pa ) such a drop (cf. Eq (1)). Because the drop is placed at the air-water interface, the difference is considered as acting in vertical direction.
Fig. 1. Amphiphilic monolayer's Langmuir system with a barrier equipped with a labile spring. The compressing pressure coming from the barrier, c , is balanced by the one of extending type, caused by van der Waals forces, e .
the water (cf. Fig. 2). The difference in the vertical pressures is related strictly to surface tension of a created drop and its charge. The drops consist of (positively) charged amphiphilic molecule with a surrounding sheath of water. After applying a barrier's oscillating load, air can intrude the monolayer in-between amphiphilic molecules creating bubbles. In the presented model, any contribution of lipids' tails is neglected as it is considered of secondary importance since they are not involved in the water's surface layer. This study aims at finding a refractive index of the vertically vs. horizontally balanced lipid monolayer system, assuming that its nonlinear dielectric response given in terms of the corresponding permittivity measure of the lipids is allowable based on the method proposed. The paper is structured as follows. In the next section (Sec. 2), we present in brief a comparative analytical method of deriving in a simple way the dielectric permittivity of Langmuir monolayers, and in Sec. 3 their optical properties are addressed [15]. The paper closes with conclusions (Sec. 4).
occupied by the molecules, k - Boltzmann's constant, and T = const. as above. e is a lateral pressure which tries to vertically extend the drop. The pressure is caused by van der Waals forces and it is related to coefficient a in the particle-particle pairwise interaction (a is of very small value because the interactions are assumed to be very weak). In order to determine the binary interaction parameter a, a Hamaker constant can be used ( AH = 2a (N /S ) 2 ). The Hamaker constant is characteristic of the Casimir effect [13]. c is just the pressure which is trying to compress the drop. This pressure is caused by the barrier which contributes with a weak lateral force as to produce a higher compactness of the system than in a situation without the barrier. In various papers on lipid monolayers, authors suggest that the monolayer at air-water interface is not continuous and consists of lipid clusters which coalescence into large ones after putting more lateral pressure on the system [18,19]. This coalescence is mediated by the water structure (see, Chapt 2.9 in Refs. [2,20]). This is in accordance with the model proposed, but in our case the pressure is weak enough to assure the possibility of getting the local equilibrium state. This scenario is possible because the lateral pressure values needed for coalescence ranged between 1 and 50 mN/m [1,19,21]. We have opted here for the minimalistic lateral pressure involving model, thus, recommending the values close to the minimum of the range suggested by literature data, i.e. quite close to 1 5 mN/m [19,21]. To equilibrate or match the two types of pressures: classical (Eq. (1)) and lateral (Eq. (2)), one may propose to compare them according to their dimensional and physical contents, i.e.
2. Method of deriving the dielectric permittivity of the Langmuir monolayer Let us describe the Langmuir lipid monolayer as a thermodynamic system at equilibrium ( µ1 (T , pw ) = µ 2 (T , pa ) ), so in saturated-vapor conditions; μ stands for the chemical potential, and the subscripts denote water and air, respectively (p and T are pressure and temperature, respectively). According to a Kubo representation of the Kelvin's formula for one charged aqueous drop (N = 1; see, Fig. 2) consisting of amphiphile's charged head, upon postponing the hydrophobic tail(s), it can be written that:
pw
2 pa = p = r
(Ze )2 1 8 0r4
N r
1
,
c
e
=
=
NkT S
S2
where N is a number of drops/aqueous spheres (N
(3)
where N r is a length of hemispheres (half-spheres, but here: semicircles, due to dimension restriction) along which the lateral pressure is applied. The length, as measured at N hemispherical caps (cf, Fig. 2), plays also a role of adjusting the pressures' physical dimensions, thus arriving at [N / m2] ultimately. The product of N p assumes an equipartition of the pressure-delivered energy absorbed by the system (uniform distribution of energy per each of N charged drops). By putting Eq. (1) and Eq. (2) into Eq. (3) one can obtain
(1)
where pw and pa stand for pressures acting from water and air (the pressures are measured in a unit of N /m2 = Pa ), γ is a surface tension, Z - number of elementary charges e, r - radius of the charged drop, ε relative electric permittivity, 0 - vacuum permittivity (see, Chapt. 4 in Ref. [16]). The second part of the right-hand side of Eq. (1) involves the classical Born correction of size originating from the energy of the polarization interaction of electron with atoms [17]. On the other hand, in Langmuir thin films’ experiments, one applies the van der Waals (modified1) state equation with weakly connected (bio)molecules [2,3]. Applying the lateral surface pressure concept, [N/m], one can obtain
aN 2
= N p,
kT 1 r S
2N Na 1 = r S2 r
N (Ze )2 1 8 0r4
1
.
(4)
Comparing then the left- and the right-hand side of Eq. (4) part by part, two separate equations appear consequently, associated with, respectively: 1/ S -part and 1/ S 2 -part. After a simple algebraic operation, they can be written as
(2)
1), S is an area
2 N 1 = , S kT
1
Modification relies on substituting the volume of a system by its surface area, denoted here by S.
and 2
(5)
Optical Materials 93 (2019) 1–5
A. Gadomski, et al. 1
(Ze )2 1 1 = S2 8a 0 r 3
.
(6)
By taking the square of Eq.(5), comparing adequately side by side Eqs (5) and (6) and solving then for this set of equations, one can obtain a quadratic equation for ε 2
(7)
+ 1 = 0.
The ε-independent part of the equation above, denoted by mensionless and can be written as
= 24 vs a
N ZekT
0
2
,
is di-
(8)
where vs = 4/3 is an eigenvolume of the charged sphere. The can be named as a collective dielectric-optical parameter. Note, that be1)/ 2 , its value has to be greater than zero, i.e. >0 . If ϕ cause = ( = 0 then ε = 1 (valid for vacuum) is naturally recovered. Solving Eq. (7) is very important from an optical point of view because dielectric permittivity, ε, is directly related to the refractive index, n [11]. From Maxwell's equations, it can be derived that
r3
Fig. 3. Refractive index n as a function of
[10]). The same value has been obtained for myelin [23]. The discrepancy between experimental and theoretical variables of n is small and points out that the presented framework works in a proper range of values which are natural for refractive indices for lipid monolayers. Note, that the neglect of the lipids' tails in our approach can be one of the reasons for the obtained minor discrepancy in n-values. Let us remind here, that Eq. (11) was provided only for specific systems for which the collective parameter, , points to = 1/4 . A more realistic case is a situation where >0 , thus <1/4 . In this case, there exist two real roots of the Eq. (7) which can be given by
(9)
= n2,
when one assumes that relative magnetic permittivity (permeability) μ of the system is close to one. We wish here to underline the very role played by K.B. Blodgett in examining the optical properties of the Langmuir films studied [22]. 3. Optical properties of the Langmuir monolayer - the exact value of its refractive index, and beyond The roots of Eq. (7) can be found when using common quadratic 1± formula as = 2 , where = 1 4 is a discriminant of the quadratic equation and its sign points out if there exist one or two roots of the equation and if they are real or complex. Thus, one can consider three situations: (i) = 0 , (ii) >0 , (iii) <0 . Because point (iii) is possible to occur only for special 2D lipid polarized materials for which dielectric permittivity is a complex function, we have focused on (i) and (ii) as the refractive index of Langmuir amphiphilic monolayer is supposed to be a real (measurable) number. In the first situation = 0 , thus, it is a case where = 1/4 . There exists only one core solution of the quadratic Eq. (7) which yields
=
1 = 2. 2
1,2
=
2
(11)
This horizontal vs. perpendicular pressure-balance involving value is of fairly remarkable accordance with some experimental values which provided the refractive index for DPPC and DPPE monolayers [11] which have been presented in Table 1. In Ref. [11], the authors measured the refractive index of n 1.48. Moreover, n-values measured for fats in various biological systems provide values around 1.46 (see, Sec. Lipids and Lipoproteins in Ref.
nmol =
n
Ref.
DPPC DPPE human fat myelin theoretical
1.48 1.49 1.46 1.46 1.41
[11] [11] [10] [23] Eq. (11)
1 2
4
.
(12)
1
1 2
4
1,
(13)
(0,1/4) . where comes out from Eq. (8) and When <0 (point (iii)) the dielectric permittivity is a complex comp = 1/2 ± i (4 1)/(4 2) . It is obtained when >1/4 . number 1,2 The real part of the dielectric permittivity is a dielectric constant which describes the ability of a dielectric material to store electrical energy, whereas the imaginary part is the ability to dissipate energy (loss factor). Based on Eq. (9), a complex refractive index comp comp ncomp = nre + i nim exists, where nre = (| 1,2 | + Re ( 1,2 ))/2 ,
Table 1 Experimentally measured and theoretically derived lipid monolayers’ refractive indices. Lipid
1±
systems have the refractive index n < 1.5, only n1 = 1 1 4 seems 2 to have an important physical meaning. Moreover, bear in mind that, thanks to Eq. (8), n gets on a fully geometric-thermodynamic and electric-part involved representative value, i.e. it is a function of five decisive physical parameters (T , a , vs, Ze , ). Applying the Gladstone-Dale relation [10], the n value can split into two values n = 1 + nmol . Because n is related to the quotient of the speed of light in a vacuum and the speed of light in the Langmuir monolayer, the fractional value, nmol , is attributed to the molecular contribution of the charged lipid-water system. This value contains information about molecular density, charge, mass, binary interactions etc. (cf., Eq. (8)) at a specific temperature T [10,11]. It can be presented as
(10)
1.41.
=
Values of n1 and n2 , computed on the basis of Eqs. (12) and (9), for (0,1/4) , have been shown in Fig. 3. Because most of water-based
Based on the relation between permittivity and refractive index, Eq. (9) and Eq. (10), one can write that
n=
(0,1/4) .
comp comp comp comp 2 comp 2 nim = (| 1,2 | Re ( 1,2 ))/2 and | 1,2 | = Re ( 1,2 ) + Im ( 1,2 ) (1/4,1/2) have been presented in [15]. Solutions of the equations for Fig. 4. Because Re ( 1comp) = Re ( 2comp) and Im ( 1comp)2 = Im ( 2comp)2 there is only one complex root of the Eq. (7), namely
3
Optical Materials 93 (2019) 1–5
A. Gadomski, et al.
dependent) equilibrium conditions is capable of yielding a clear dielectric, thus, unique optical response of the system of interest. Declaration of interests None. Agreement All authors have seen and approved the final version of the manuscript being submitted. They warrant that the article is the authors' original work, hasn't received prior publication and isn't under consideration for publication elsewhere. Acknowledgements This work is supported by UTP University of Science and Technology, Poland BN-10/19. One of us (AG) is indebted to Prof. Zbigniew J. Grzywna, STU Gliwice, Poland, for his great tolerance on first assessing about thirty years ago by AG the computer-simulation based comprehension of Langmuir films with fractal domains of dimension ca. dF = 1.5. Authors also want to thank Sophie Miller (Tulane University), Fulbright ETA at UTP, for her impact as a native speaker in obtaining the final form of the paper.
Fig. 4. Complex refractive index ncomp = nre + i nim as a function of (1/4,1/2) , considered for lipid polarized materials.
ncomp =
1 4 2
+
(
4
1 4 2
)+
1 2
/2 + i
1 4 2
+
(
4
1 4 2
)
1 2
/2 ,
where the real part is a refractive index and the imaginary part is an absorption index.
References
4. Conclusions
[1] A. Ulman, Formation and structure of self-assembled monolayers, Chem. Rev. 96 (4) (1996) 15331554. [2] A. Ulman, An Introduction to Ultrathin Organic Films: from Langmuir-Blodgett to Self-Assembly, Academic Press, London, 1991. [3] A. Miller, W. Knoll, H. Möhwald, Fractal growth of crystalline phospholipid domains in monomolecular layers, Phys. Rev. Lett. 56 (1986) 2633. [4] P. Laggner, M. Kriechbaum, Phospholipid phase transitions: kinetics and structural mechanisms, Chem. Phys. Lipids 57 (1991) 121. [5] A. Gadomski, Polymorphic phase transitions in systems evolving in a two-dimensional discrete space, Phys. Rev. E 60 (1999) 1252. [6] Z. Pawlak, A. Gadomski, M. Sojka, W. Urbaniak, P. Bełdowski, The amphoteric effect on friction between the bovine cartilage/cartilage surfaces under slightly sheared hydration lubrication mode, Colloids Surfaces B Biointerfaces 146 (2016) 452458. [7] J. Siódmiak, A. Gadomski, E. Pechkova, C. Nicolini, Computer model of a lysozyme crystal growth with/without nanotemplate - a comparison, Int. J. Mod. Phys. C 17 (2006) 1359. [8] S.I. Valyansky, E.K. Naimi, L.V. Kozhitov, Functional 2D nanomaterials for optoelectronics based on Langmuir bacteriorhodopsin films, Modern Electronic Materials 2 (2016) 79–84. [9] G. Cevc, D. Marsh, Phospholipid Bilayers: Physical Principles and Models, Wiley, New York, 1987. [10] R. Barer, S. Joseph, Refractometry of living cells, J. Cell Sci. s3–95 (1954) 399–423. [11] D.F. Kienle, J.V. Souza, E.B. Watkins, T.L. Kuhl, Thickness and refractive index of DPPC and DPPE monolayers by multiple-beam interferometry, Anal. Bioanal. Chem. 406 (2014) 4725–4733. [12] J. Israelashvili, Intermolecular and Surface Forces, 3 edition, Academic Press, 2011 Chapt. 20. [13] F. Soyka, O. Zvyagolskaya, C. Hertlein, L. Helden, C. Bechinger, Critical Casimir forces in colloidal suspensions on chemically patterned surfaces, Phys. Rev. Lett. 101 (2008) 208301. [14] M. Tröndle, S. Kondrat, A. Gambassi, L. Harnau, S. Dietrich, Normal and lateral critical Casimir forces between colloids and patterned substrates, Europhys. Lett. 88 (2009) 40004. [15] D.W. Van Krevelen, Properties of Polymers. Correlations with Chemical Structure, Elsevier, Amsterdam, 1972 Chap.11. [16] R. Kubo, Thermodynamics, North-Holland Publishing Company, Amsterdam, 1968. [17] V.V. Pogosov, A. Kiejna, D.P. Kotlyarov, Sum-rule approach in the theory of charged self-compressed dielectric droplets, Phys. A 293 (2001) 59–70. [18] G. Colacicco, Ar Buckelew Jr., E.W.J. Scarpello, Lipid monolayers: surface viscosity of dipalmitoyl lecithin in relation to surface potential and ion binding, J. Colloid Interface Sci. 46 (1974) 147–151. [19] R.E. Brown, H.L. Brockman, Using monomolecular films to characterize lipid lateral interactions, Methods Mol. Biol. 398 (2007) 4158. [20] Y.L. Chen, C.A. Helm, J.N. Israelachvili, Measurements of the elastic properties of surfactant and lipid monolayers, Langmuir 7 (1991) 26942699. [21] A. Miller, H. Möhwald, Diffusion limited growth of crystalline domains in phospholipid monolayers, J. Chem. Phys. 86 (1987) 4258. [22] K.B. Blodgett, I. Langmuir, Built-up films of barium stearate and their optical properties, Phys. Rev. 51 (1937) 964–982.
It is important to study relevant, effectively two-dimensional biomolecular systems [24] for which a reduction of the third degree of freedom and some controlled utilization of the rules of mesoscopic nonequilibrium thermodynamics allow for both their better control [25] and efficient usage [7]. It turns out that the simple derivation of lipids' monolayer refractive index demonstrates meaningful practical connotations. This is because so many pronounced physicochemical scenarios unite at the close-toequilibrium local thermodynamic conditions at some respective value of temperature of the system. Firstly, two distinct scenarios, that of Langmuir with Van der Walls isotherm, and the Kelvin-Laplace (electric spherical cloud involving) uniform pressure, are actually related. It is plausible for them both to merge readily at a specific pressure-temperature critical point for which another scenario, namely that of Gladstone-Dale [10,15] tends to apply. It is to be speculated that the fractional residuum of the irrational value of the lipids' refractive index, n 2 1.41, namely the value of ca. 0.41, contains a meaningful message about the molar molecular density of the lipids [10], which are very much related with lipidic system's binary interactions, thus with the Hamaker's constant of the lipids [11,12]. Let us recall again, that the Hamaker constant describes the Casimir-type effect [13,14], provided by the weakly acting barrier (cf., Fig. 1), which provides critical fluctuations into the system, assuring the probability of creating small charged drops of lipids with sheath of water at the air-water interface. Of course, such a rationale presented throughout the paper ought to be confronted with experiments [2,9–11,20,26]. But the confrontation 2 yields the thermolooks promising since our derivation with n dynamic-theory based value close to those characteristics of lipidic systems [1,10,11]. This refers to a high motivation of presenting the underlying study to a critical reader looking to rationalize the values of refractive indices for optical thin-film viscoelastic/lipidic (nano)materials [8,27] for which certain efficient measurements have been performed in the past [20]. Finally, it is worth underlining that a key novelty of our simplistic approach of getting nonlinear optical formulae for the in-plane built lipids by emphasizing the fact that the thermodynamic procedure based on equations of state(s) and valid at (local: pressure and temperature 4
Optical Materials 93 (2019) 1–5
A. Gadomski, et al. [23] J.M. Pusterla, A.A. Malfatti-Gasperini, X.E. Puentes-Martinez, L.P. Cavalcanti, R.G. Oliveira, Refractive index and thickness determination in Langmuir monolayers of myelin lipids, Biochim. Biophys. Acta 1859 (2017) 924930. [24] A. Gadomski, J. Luczka, M. Kriechbaum, P. Laggner, A. Jamnik, Diffusion-migration concept applied to the formation of microdomains in model 2D-biomaterials, Phys. Lett. 203 (1995) 367. [25] A. Gadomski, N. Kruszewska, M. Ausloos, J. Tadych, On the harmonic mean property of model dispersive systems emerging under mononuclear, mixed and
polynuclear path conditions, in: K. Schadschneider, et al. (Ed.), Springer Lecture Notes in Applied Mathematics, 2007, pp. 283–296 Part II Transport in Biological Systems. [26] P. Kościelniak, A. Grzeszczak, J. Szuber, XPS and AFM studies of surface chemistry and morphology of In2O3 ultrathin films deposited by rheotaxial growth and vacuum oxidation after air exposure, Cryst. Res. Technol. 50 (2015) 884–890. [27] M. Radmacher, R.W. Tillmann, H.E. Gaub, Imaging viscoelasticity by force modulation with the atomic force microscope, Biophys. J. 64 (1993) 735–742.
5
Contents lists available at ScienceDirect
Optical Materials journal homepage: www.elsevier.com/locate/optmat
Derivation of the refractive index of lipid monolayers at an air-water interface
T
A. Gadomskia, N. Kruszewskaa,∗, J.M. Rubib a b
UTP University of Science and Technology, Institute of Mathematics and Physics, Kaliskiego 7, PL–85796, Bydgoszcz, Poland Universitat de Barcelona, Departament de Fisica de la Materia Condensada, Diagonal 647, 08028, Barcelona, Spain
ARTICLE INFO
ABSTRACT
Keywords: Langmuir thin film Soft optical materials Van der Waals type equation of state Kelvin-Kubo law for dielectric spherical cloud Refractive index Lipids
We demonstrate by means of a simple thought experiment on lipid films dispersed over an air-water interface that a core refractive index of the system approaches 2 , a very significant value for the lipid (DPPC or DPPE) monolayers. The thought experiment consists of balancing two principal types of acting pressures: the vertical and the horizontal/lateral ones, both taken at a thermodynamic equilibrium. The horizontal pressure of (modified) van der Waals type comes from the minimalistic application of the (movable) lateral barrier, as can be expected to occur in Langmuir experiments, provided that the application of pressure is very weak. On the other hand, after relaxing the system when the barrier goes back, one expects to arrive at another scenario. This is when the vertical pressure of the air-water quasi-planar system of hydrophobic propensity applies, provided that an idealized assumption of the equally distributed pressurizing energy is true. This is in accord with a Kubo evaluation for a certain number of charged point-like objects immersed in water spherical shells and surrounded by air for which the Kelvin (J.J. Thomson) law prevails. At the horizontal vs. vertical pressure conditions one may uncover, by employing simple analytic means the basic dielectric and optical properties of the domain-wise, pairwise-interaction involving lipid film. It turns out that a simple Gladstone-Dale scenario, pointing to splitting 1 + 0.41 overwhelms, with the fractional value of ca. 0.41 attributed to a the core refractive index into 2 molecular contribution of the charged lipid-water (dipolar) local system, affecting the overall pressure-addressing scenario. (By the value of refractive index centering at one, the vacuum conditions are to be addressed.) One may argue, however, that the fractional part of the refractive index seems to be a bit outside the contemporary experimental reach. On the other hand, it can be rationalized by the Casimir critical soft-matter fluctuational effect.
1. Introduction Lipid monolayers dispersed over an air-water interface are widely accepted as an interesting model system prone to a plethora of phase changes, resulting in an emergence of microdomains in its compressed lipid phase submerged in water [1–6]. There are many applications of lipid systems with versatile physicochemical characteristics, especially in biosensoring, thin biofilms’ formation, bionanotechnology, pharmacology, biomaterials engineering and biomolecular crystal technology, and the likes [2,7,8]. There is also a plausible extension of these applications toward biophysics and cell biology, and in particular model biomembrane formation, with its kinetics and thermodynamics that are of vital biomedical interest [9–11]. In this short communication, we consider an effectively two-
∗
dimensional system at the air-water interface. The system is an amphiphilic (viz lipid) monolayer thin film of Langmuir type (cf. Fig. 1) [12]. In order to achieve the attraction between the amphiphilic molecules’ heads by getting them closer to each other, a weak (oscillating) pressure, provided by a barrier, is applied. The barrier is equipped with a labile spring with a very small spring constant (suitable for standardtemperature and/or close-to-physiologic conditions). This way, there is a chance for amphiphilic molecules to get (locally) closer to each other, and thanks to Casimir-type effect [13,14], providing critical fluctuations in the system, to eventually create small drops or microdomains. The (compressing) pressure coming from the barrier is balanced by the pressure of the extending type, caused by van der Waals forces. Vectors of these two (compressing and extending) forces have a horizontal direction (cf., Fig. 1). In vertical direction, there are also two competing pressures: the one coming from the air and the second - from
Corresponding author. E-mail address: [email protected] (N. Kruszewska).
https://doi.org/10.1016/j.optmat.2019.04.042 Received 30 October 2018; Received in revised form 15 April 2019; Accepted 20 April 2019 Available online 10 May 2019 0925-3467/ © 2019 Elsevier B.V. All rights reserved.
Optical Materials 93 (2019) 1–5
A. Gadomski, et al.
Fig. 2. Simplified sketch of a drop (charged sphere) of radius r half-submerged in a water. The drop consists of one charged amphiphilic molecule's head surrounded by water molecules. Based on Kelvin's equation there is a difference between pressures inside ( pw ) and outside ( pa ) such a drop (cf. Eq (1)). Because the drop is placed at the air-water interface, the difference is considered as acting in vertical direction.
Fig. 1. Amphiphilic monolayer's Langmuir system with a barrier equipped with a labile spring. The compressing pressure coming from the barrier, c , is balanced by the one of extending type, caused by van der Waals forces, e .
the water (cf. Fig. 2). The difference in the vertical pressures is related strictly to surface tension of a created drop and its charge. The drops consist of (positively) charged amphiphilic molecule with a surrounding sheath of water. After applying a barrier's oscillating load, air can intrude the monolayer in-between amphiphilic molecules creating bubbles. In the presented model, any contribution of lipids' tails is neglected as it is considered of secondary importance since they are not involved in the water's surface layer. This study aims at finding a refractive index of the vertically vs. horizontally balanced lipid monolayer system, assuming that its nonlinear dielectric response given in terms of the corresponding permittivity measure of the lipids is allowable based on the method proposed. The paper is structured as follows. In the next section (Sec. 2), we present in brief a comparative analytical method of deriving in a simple way the dielectric permittivity of Langmuir monolayers, and in Sec. 3 their optical properties are addressed [15]. The paper closes with conclusions (Sec. 4).
occupied by the molecules, k - Boltzmann's constant, and T = const. as above. e is a lateral pressure which tries to vertically extend the drop. The pressure is caused by van der Waals forces and it is related to coefficient a in the particle-particle pairwise interaction (a is of very small value because the interactions are assumed to be very weak). In order to determine the binary interaction parameter a, a Hamaker constant can be used ( AH = 2a (N /S ) 2 ). The Hamaker constant is characteristic of the Casimir effect [13]. c is just the pressure which is trying to compress the drop. This pressure is caused by the barrier which contributes with a weak lateral force as to produce a higher compactness of the system than in a situation without the barrier. In various papers on lipid monolayers, authors suggest that the monolayer at air-water interface is not continuous and consists of lipid clusters which coalescence into large ones after putting more lateral pressure on the system [18,19]. This coalescence is mediated by the water structure (see, Chapt 2.9 in Refs. [2,20]). This is in accordance with the model proposed, but in our case the pressure is weak enough to assure the possibility of getting the local equilibrium state. This scenario is possible because the lateral pressure values needed for coalescence ranged between 1 and 50 mN/m [1,19,21]. We have opted here for the minimalistic lateral pressure involving model, thus, recommending the values close to the minimum of the range suggested by literature data, i.e. quite close to 1 5 mN/m [19,21]. To equilibrate or match the two types of pressures: classical (Eq. (1)) and lateral (Eq. (2)), one may propose to compare them according to their dimensional and physical contents, i.e.
2. Method of deriving the dielectric permittivity of the Langmuir monolayer Let us describe the Langmuir lipid monolayer as a thermodynamic system at equilibrium ( µ1 (T , pw ) = µ 2 (T , pa ) ), so in saturated-vapor conditions; μ stands for the chemical potential, and the subscripts denote water and air, respectively (p and T are pressure and temperature, respectively). According to a Kubo representation of the Kelvin's formula for one charged aqueous drop (N = 1; see, Fig. 2) consisting of amphiphile's charged head, upon postponing the hydrophobic tail(s), it can be written that:
pw
2 pa = p = r
(Ze )2 1 8 0r4
N r
1
,
c
e
=
=
NkT S
S2
where N is a number of drops/aqueous spheres (N
(3)
where N r is a length of hemispheres (half-spheres, but here: semicircles, due to dimension restriction) along which the lateral pressure is applied. The length, as measured at N hemispherical caps (cf, Fig. 2), plays also a role of adjusting the pressures' physical dimensions, thus arriving at [N / m2] ultimately. The product of N p assumes an equipartition of the pressure-delivered energy absorbed by the system (uniform distribution of energy per each of N charged drops). By putting Eq. (1) and Eq. (2) into Eq. (3) one can obtain
(1)
where pw and pa stand for pressures acting from water and air (the pressures are measured in a unit of N /m2 = Pa ), γ is a surface tension, Z - number of elementary charges e, r - radius of the charged drop, ε relative electric permittivity, 0 - vacuum permittivity (see, Chapt. 4 in Ref. [16]). The second part of the right-hand side of Eq. (1) involves the classical Born correction of size originating from the energy of the polarization interaction of electron with atoms [17]. On the other hand, in Langmuir thin films’ experiments, one applies the van der Waals (modified1) state equation with weakly connected (bio)molecules [2,3]. Applying the lateral surface pressure concept, [N/m], one can obtain
aN 2
= N p,
kT 1 r S
2N Na 1 = r S2 r
N (Ze )2 1 8 0r4
1
.
(4)
Comparing then the left- and the right-hand side of Eq. (4) part by part, two separate equations appear consequently, associated with, respectively: 1/ S -part and 1/ S 2 -part. After a simple algebraic operation, they can be written as
(2)
1), S is an area
2 N 1 = , S kT
1
Modification relies on substituting the volume of a system by its surface area, denoted here by S.
and 2
(5)
Optical Materials 93 (2019) 1–5
A. Gadomski, et al. 1
(Ze )2 1 1 = S2 8a 0 r 3
.
(6)
By taking the square of Eq.(5), comparing adequately side by side Eqs (5) and (6) and solving then for this set of equations, one can obtain a quadratic equation for ε 2
(7)
+ 1 = 0.
The ε-independent part of the equation above, denoted by mensionless and can be written as
= 24 vs a
N ZekT
0
2
,
is di-
(8)
where vs = 4/3 is an eigenvolume of the charged sphere. The can be named as a collective dielectric-optical parameter. Note, that be1)/ 2 , its value has to be greater than zero, i.e. >0 . If ϕ cause = ( = 0 then ε = 1 (valid for vacuum) is naturally recovered. Solving Eq. (7) is very important from an optical point of view because dielectric permittivity, ε, is directly related to the refractive index, n [11]. From Maxwell's equations, it can be derived that
r3
Fig. 3. Refractive index n as a function of
[10]). The same value has been obtained for myelin [23]. The discrepancy between experimental and theoretical variables of n is small and points out that the presented framework works in a proper range of values which are natural for refractive indices for lipid monolayers. Note, that the neglect of the lipids' tails in our approach can be one of the reasons for the obtained minor discrepancy in n-values. Let us remind here, that Eq. (11) was provided only for specific systems for which the collective parameter, , points to = 1/4 . A more realistic case is a situation where >0 , thus <1/4 . In this case, there exist two real roots of the Eq. (7) which can be given by
(9)
= n2,
when one assumes that relative magnetic permittivity (permeability) μ of the system is close to one. We wish here to underline the very role played by K.B. Blodgett in examining the optical properties of the Langmuir films studied [22]. 3. Optical properties of the Langmuir monolayer - the exact value of its refractive index, and beyond The roots of Eq. (7) can be found when using common quadratic 1± formula as = 2 , where = 1 4 is a discriminant of the quadratic equation and its sign points out if there exist one or two roots of the equation and if they are real or complex. Thus, one can consider three situations: (i) = 0 , (ii) >0 , (iii) <0 . Because point (iii) is possible to occur only for special 2D lipid polarized materials for which dielectric permittivity is a complex function, we have focused on (i) and (ii) as the refractive index of Langmuir amphiphilic monolayer is supposed to be a real (measurable) number. In the first situation = 0 , thus, it is a case where = 1/4 . There exists only one core solution of the quadratic Eq. (7) which yields
=
1 = 2. 2
1,2
=
2
(11)
This horizontal vs. perpendicular pressure-balance involving value is of fairly remarkable accordance with some experimental values which provided the refractive index for DPPC and DPPE monolayers [11] which have been presented in Table 1. In Ref. [11], the authors measured the refractive index of n 1.48. Moreover, n-values measured for fats in various biological systems provide values around 1.46 (see, Sec. Lipids and Lipoproteins in Ref.
nmol =
n
Ref.
DPPC DPPE human fat myelin theoretical
1.48 1.49 1.46 1.46 1.41
[11] [11] [10] [23] Eq. (11)
1 2
4
.
(12)
1
1 2
4
1,
(13)
(0,1/4) . where comes out from Eq. (8) and When <0 (point (iii)) the dielectric permittivity is a complex comp = 1/2 ± i (4 1)/(4 2) . It is obtained when >1/4 . number 1,2 The real part of the dielectric permittivity is a dielectric constant which describes the ability of a dielectric material to store electrical energy, whereas the imaginary part is the ability to dissipate energy (loss factor). Based on Eq. (9), a complex refractive index comp comp ncomp = nre + i nim exists, where nre = (| 1,2 | + Re ( 1,2 ))/2 ,
Table 1 Experimentally measured and theoretically derived lipid monolayers’ refractive indices. Lipid
1±
systems have the refractive index n < 1.5, only n1 = 1 1 4 seems 2 to have an important physical meaning. Moreover, bear in mind that, thanks to Eq. (8), n gets on a fully geometric-thermodynamic and electric-part involved representative value, i.e. it is a function of five decisive physical parameters (T , a , vs, Ze , ). Applying the Gladstone-Dale relation [10], the n value can split into two values n = 1 + nmol . Because n is related to the quotient of the speed of light in a vacuum and the speed of light in the Langmuir monolayer, the fractional value, nmol , is attributed to the molecular contribution of the charged lipid-water system. This value contains information about molecular density, charge, mass, binary interactions etc. (cf., Eq. (8)) at a specific temperature T [10,11]. It can be presented as
(10)
1.41.
=
Values of n1 and n2 , computed on the basis of Eqs. (12) and (9), for (0,1/4) , have been shown in Fig. 3. Because most of water-based
Based on the relation between permittivity and refractive index, Eq. (9) and Eq. (10), one can write that
n=
(0,1/4) .
comp comp comp comp 2 comp 2 nim = (| 1,2 | Re ( 1,2 ))/2 and | 1,2 | = Re ( 1,2 ) + Im ( 1,2 ) (1/4,1/2) have been presented in [15]. Solutions of the equations for Fig. 4. Because Re ( 1comp) = Re ( 2comp) and Im ( 1comp)2 = Im ( 2comp)2 there is only one complex root of the Eq. (7), namely
3
Optical Materials 93 (2019) 1–5
A. Gadomski, et al.
dependent) equilibrium conditions is capable of yielding a clear dielectric, thus, unique optical response of the system of interest. Declaration of interests None. Agreement All authors have seen and approved the final version of the manuscript being submitted. They warrant that the article is the authors' original work, hasn't received prior publication and isn't under consideration for publication elsewhere. Acknowledgements This work is supported by UTP University of Science and Technology, Poland BN-10/19. One of us (AG) is indebted to Prof. Zbigniew J. Grzywna, STU Gliwice, Poland, for his great tolerance on first assessing about thirty years ago by AG the computer-simulation based comprehension of Langmuir films with fractal domains of dimension ca. dF = 1.5. Authors also want to thank Sophie Miller (Tulane University), Fulbright ETA at UTP, for her impact as a native speaker in obtaining the final form of the paper.
Fig. 4. Complex refractive index ncomp = nre + i nim as a function of (1/4,1/2) , considered for lipid polarized materials.
ncomp =
1 4 2
+
(
4
1 4 2
)+
1 2
/2 + i
1 4 2
+
(
4
1 4 2
)
1 2
/2 ,
where the real part is a refractive index and the imaginary part is an absorption index.
References
4. Conclusions
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It is important to study relevant, effectively two-dimensional biomolecular systems [24] for which a reduction of the third degree of freedom and some controlled utilization of the rules of mesoscopic nonequilibrium thermodynamics allow for both their better control [25] and efficient usage [7]. It turns out that the simple derivation of lipids' monolayer refractive index demonstrates meaningful practical connotations. This is because so many pronounced physicochemical scenarios unite at the close-toequilibrium local thermodynamic conditions at some respective value of temperature of the system. Firstly, two distinct scenarios, that of Langmuir with Van der Walls isotherm, and the Kelvin-Laplace (electric spherical cloud involving) uniform pressure, are actually related. It is plausible for them both to merge readily at a specific pressure-temperature critical point for which another scenario, namely that of Gladstone-Dale [10,15] tends to apply. It is to be speculated that the fractional residuum of the irrational value of the lipids' refractive index, n 2 1.41, namely the value of ca. 0.41, contains a meaningful message about the molar molecular density of the lipids [10], which are very much related with lipidic system's binary interactions, thus with the Hamaker's constant of the lipids [11,12]. Let us recall again, that the Hamaker constant describes the Casimir-type effect [13,14], provided by the weakly acting barrier (cf., Fig. 1), which provides critical fluctuations into the system, assuring the probability of creating small charged drops of lipids with sheath of water at the air-water interface. Of course, such a rationale presented throughout the paper ought to be confronted with experiments [2,9–11,20,26]. But the confrontation 2 yields the thermolooks promising since our derivation with n dynamic-theory based value close to those characteristics of lipidic systems [1,10,11]. This refers to a high motivation of presenting the underlying study to a critical reader looking to rationalize the values of refractive indices for optical thin-film viscoelastic/lipidic (nano)materials [8,27] for which certain efficient measurements have been performed in the past [20]. Finally, it is worth underlining that a key novelty of our simplistic approach of getting nonlinear optical formulae for the in-plane built lipids by emphasizing the fact that the thermodynamic procedure based on equations of state(s) and valid at (local: pressure and temperature 4
Optical Materials 93 (2019) 1–5
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