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Shape Fluctuations of Free Liquid Films with Diffuse Electric Double LayersB. U. Felderhof Citation: The Journal of Chemical Physics 48, 1178 (1968); doi: 10.1063/1.1668779 View online: http://dx.doi.org/10.1063/1.1668779 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/48/3?ver=pdfcov Published by the AIP Publishing Articles you may be interested in The tail effect on the shape of an electrical double layer differential capacitance curve J. Chem. Phys. 138, 144704 (2013); 10.1063/1.4799886 Diffuse double layer charging in nonpolar liquids Appl. Phys. Lett. 91, 182911 (2007); 10.1063/1.2805229 High source potential upstream of a current-free electric double layer Phys. Plasmas 12, 044508 (2005); 10.1063/1.1883182 Erratum: Theory of the Diffuse Double Layer in Liquids J. Chem. Phys. 41, 2554 (1964); 10.1063/1.1726304 Theory of the Diffuse Double Layer in Liquids J. Chem. Phys. 40, 628 (1964); 10.1063/1.1725182
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THE JOURNAL OF CHEMICAL PHYSICS VOLUME 48, NUMBER 3 1 FEBRUARY 1968
Shape Fluctuations of Free Liquid Films with Diffuse Electric Double Layers
B. U. FELDERHOF
Institutefor Theoretical Physics, Rijksuniversiteit, Utrecht, the Netherlands
(Received 13 July 1967)
The fluctuations in shape of free liquid films which may be-observed by light scattering yield informati~n about the forces determining film stability. The main forces are the van der ~aal~ force ~nd the elec~nc force caused by the overlap of diffuse electric double layers f?rmed by the lOns I~ solutI~n. ': detailed analysis employing the thermodynamic theory of fluctuations IS made o~ the electn.c contnbutio~ to the fluctuations in shape. The thermodynamics of the double-layer system IS kept as sImple as pOSSIble but account is taken of the nature of ion adsorption at the film surface.
1. INTRODUCTION
The study of the stability of thin liquid films is of interest both from a practical and a purely scientific point of view. A well-known example of a free liquid lamella is the soap film; also, in biology, liquid lamellae occur in many instances.1 The structure of such films may briefly be described as consisting of a volume part bounded by surface layers. The volume part has a thickness of a few hundred angstroms and contains an electrolyte solution. The surface layers with a thick~ess of some 10 A consist of adsorbed molecules and lOns and have a complicated structure.
It is now well established that the main forces of interaction determining the stability of a free film are the attractive van der Waals force and the disjoining pressure caused by the overlap of diffuse electric double layers formed at the surfaces by the ions in solution.2-5 These are the same forces as those that act between colloidal particles in solution and therefore the study of the stability of free liquid films is also of great importance for colloid chemistry.
The first direct measurements of these forces were made by Derjaguin and his school in the Soviet Un~on and their experiments were followed by exte~s~ve research elsewhere. A more indirect way of obtammg information on the forces was developed by Vrij6 who performed light scattering experiments on soap films. The main contribution to the scattering arises from the small thermal corrugations of the surface, the scattering from the bulk being small in comparison. As the scattering is proportional to the mean-square amplitude of
I For comprehensive reviews see J. Th. G. Overbeek, J. Phys. Chem. 64, 1178 (1960); J. A. Kitchener, E~deavour. 22, 118 (1963). An extensive ~eview is given?y). A. Kltchener, III Recent Progress in Surface Sc~ence, J. F. Damelh, K. G. A. Pankhurst, and A. C. Riddiford, Eds. (Academic Press Inc., New York, 1964), Vo!' 1, p. 51.
2 I. Langmuir, J. Chern. Phys. 6, 893 (1938). 3 B. V. Derjaguin, Trans. Faraday ~oc. ~6, 203 (1940); B. V.
Derjaguin and L. Landau, Acta PhyslCochlm. U.R.S.S. 14, 633 (1941). bu'
4 E. J. W. Verwey andJ. Th. G. Overbeek, Theory of the Sta tty of Lyophobic Colloids (Elsevier Pub!. Co: Inc., New YO.rk, 1948).
6 D. A. Haydon, in Recent Progress tn Surface SCtence, J. l!'. Danielli, K. G. A. Pankhurst, and A. C. Riddiford, Eds. (AcademIc Press Inc., New York, 1964), Vol. 1, p. 94.
6 A. Vrij, J. Colloid Sci. 19, 1 (1964).
the surface waves one obtains information on the forces determining the stability of the film.
The contribution of the van der Waals force to the surface fluctuations is found by a straightforward calculation and will not be considered here. Our purpose is to investigate the contribution of the disjoining pressure and the way this is influenced by adsorption in the surface layers. We take a macroscopic point of view and use thermodynamic fluctuation theory. Also we adopt a simple model in which the thickness ~f t~e surface layers is neglected, the electrolyte solutlOn IS
dilute and symmetric, and there is only adsorption ~f the soluble ions. The structure of the surface layers IS
embodied in the surface equation of state which is assumed known but need not be specified. We omit refinements from the theory such as the ionic volume effect,7 image-self-atmosphere effects, and the discret.eness-of-charge effect,8.9 in the expectation that the mam features of the problem show up in the treatment of a simple case.
In the next section the problem is outlined in more detail. Sections III-VI deal with the thermodynamics of plane-parallel double layers which is closely related to that developed by Derjaguin and Landau3 and by Verwey and Overbeek4 for the stability of lyophobic colloids. The influence of the adsorbing layers is treated in more detail however and our formulation accordingly is slightly different in that charging processes are avoided. Sections VII-IX deal more explicitly with the fluctuations. In the last section we sum up the conclusions to be drawn from the present investigation.
II. OUTLINE OF THE PROBLEM
The following simple model of a free liquid film is considered. The film consists of three parts, a volume part sandwiched between two surface laye~s. The volume part contains a dilute electrolyte solutlOn; for simplicity we consider uni-un~valent ele~trolyt~s. ~he solvent will be taken to be an mcompresslble flUld WIth
7 S. Levine and G. M. Bell, J. Phys. Chern. 64, 1188 (1960). 8 F. P. Buff and F. H. Stillinger, J. Chern. Phys. 39, 1911. (1963). 9 G. M. Bell and S. Levine, Z. Physik. Chern. (LeipZIg) 231,
289 (1966).
1178
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FLUCTUATIONS OF FREE LIQUID FILMS 1179
a large specific heat so that only isothermal processes need be considered. Both cations and anions may be adsorbed in the surface layers and we assume there are no insoluble ions. The detailed structure of the surface layers will be disregarded and we shall idealize them as infinitesimally thin two-dimensional thermodynamic systems located at a mathematical surface S bounding the volume part. The location of the surface S must be identified with the so-called outer Helmholtz plane since by definition this is the boundary of the diffuse layer in the volume part.10
It will be assumed that the surface equation of state is known, i.e., for given temperature the free energy per unit area of surface layer is a known function of the numbers of adsorbed cations and anions. Since either cations or anions are the more easily adsorbed on S there results in general a surface charge density and, consequently, a diffuse electric double-layer system is built up in the bulk. It is convenient to write the total ion contribution to the free energy of the film as a sum of two terms
(2.1)
where 5'. is the free energy of the surface phase which depends only on the distribution of adsorbed ions over the surface, and 5'd is the double-layer free energy which depends on the surface charge density and on the distribution of ions in the volume phase. Explicit definitions will be given in the following sections.
So far the shape of the film has not been specified. For a given shape and for given numbers of cations and anions in the film if thermal equilibrium is reached the distribution of ions over volume and surface is such that the total ion free energy is minimal. In order to find the probability distribution of small deviations from a completely flat shape the ion free energies for different shapes must be compared. The most convenient way of doing this is to divide a film of given shape into flat homogeneous elements, i.e., film elements of constant thickness whose properties do not vary along the film. It will be shown that such flat elements obey a rather simple thermodynamics. If the flat elements are sufficiently large and edge effects between elements may be neglected the total ion free energy may to a good approximation be written as a sum over elements.
The main complication we have to deal with in the present problem is that, even if the film in complete equilibrium is neutral, when it deviates by thermal fluctuations from its flat shape the different film elements in general have a nonvanishing net charge. This gives rise to electrostatic interaction between different film elements. In the approximation where edge effects between film elements may be neglected the total ion
10 D. C. Graharne, Chern. Rev. 41,441 (1947).
free energy for a given shape is given by (2.1) with
5',= LF',h i
(2.2)
where the sum runs over all elements. F.,; is the surface free energy of element j which depends only on the numbers of adsorbed ions. F • .; is a volume free energy to be defined which depends only on the number of ions in the volume part of elementj and on its thickness hj. The last term in (2.2) is the electrostatic Coulomb energy arising from the net charges of different film elements. This term will be called the outside field energy. The net charge Qj is the sum of surface and volume charge in element j. If the total number of ions is kept fixed the total charge LjQj vanishes. <1» is the value of the electrostatic potential at the surface of element j which, as will be shown, depends only on the net charges of all elements.
IIT. ION DISTRIBUTION IN THE VOLUME PART
It is convenient first to consider the double-layer system separately and determine its free energy for a given distribution of ions over the surface. Let n+.(s) and n-a(s) be the number of cations and anions per unit area at the point s of the surface S. The surface charge density <res) is defined by
O'(s) =e[n+-(s) -n-o(s)], (3.1)
where e is the elementary charge. Let n+(r) and n_(r) be the number of cations and anions per unit volume at point r of the volume part. The volume charge density r(r) is defined by
r(r) =e[n+(r) -n_(r)]. (3.2)
We define the double-layer free energy of the film as the sum of entropy terms for the ions in solution and the total Coulomb energy,
5'd=kT 1'0 [n+(r) logn+(r) -n+(r)
+n_(r) logn_(r) -n_(r) ]dar
+ i { r(r) .p(r)dar+ i fs <r( s).p( s) d2s, (3.3)
where kT is Boltzmann's constant times absolute temperature, and '0 is the film volume of given shape. The electrostatic potential .p(r) for a solvent of dielectric constant e is given by the Poisson integrals
1 T(r') , 1 O'(S) .p(r) = I ' I dar + I I d2s; "() e r-r s e r-s
(3.4)
.p( s) is the value of this potential at a point s of the surface S. The integrals run over the film volume and
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1180 B. U. FELDERHOF
over the film surface, respectively. The relation between Expression (3.3) for the double-layer free energy with expressions occurring in the literature 4,11,12 will be given in Sec. IV.
For fixed surface charge density 0"(8) and given numbers of ions in the volume part,
Iv n_(r)dar=<JL., (3.5)
the number densities n+(r) and n_(r) of cations and anions in the bulk adjust themselves in such a way that the total double-layer free energy (fd, defined by (3.3), is minimal. If the first variation1a of (fd is formed for variations on+(r) and on_(r) in the bulk number densities one obtains
O(fd= Iv {[kT logn+(r) +e4>(r) ]on+(r)
+[kT logn_(r) -e4>(r) ]on_(r) } dar. (3.6)
The conditions that (fd be minimal under the subsidiary condition that :J4. and <JL. are constant, which implies
f'I.J on+(r)dar=O,
are therefore
kT logn+(r) +e4>(r) =11+,
(3.7)
kT logn_(r) -e4>(r) =/l-, (3.8)
where J.I+ and J.I- are the electrochemical potentials for cations and anions. These are determined up to an additive constant which is a function of temperature only. The choice of reference potentials in (3.8) has been made as simple as possible; a similar choice of constants is implicit in (3.3).
From (3.4) and (3.8) it follows that 4>(r) satisfies a Poisson-Boltzmann equation. It is convenient to subtract a constant from 4>(r) and define
and I/;(r) satisfies the familiar Poisson-Boltzmann equation
EAI/;-47rno({exp(et/;/kT) -exp( -et/;/kT)] =0. (3.13)
In the case of film geometry the surface consists of two parts St and S II the upper and lower surface, respectively. If the film is divided into flat homogeneous elements, as defined in Sec. II, the surface charge densities O"t and 0" ~ in each element are constant. For our purpose it suffices to consider the similar case where O"t=O" +=0". For given numbers N+v and N_. of ions in the bulk of the element the minimum contribution to the free energy is attained for a solution I/;(z') of the Poisson-Boltzmann equation (3.13) which depends only on the coordinate z' perpendicular to the film element and which is symmetric with respect to the midplane z' =0. Solutions of this form14,15 constitute a two-parameter family with parameters no and 1/;(0) which are uniquely determined by N +. and N -v' Thus the bulk number densities in the element are completely determined according to (3.12).
The remaining constant 'Po is most easily expressed in terms of 1/;0, the value of I/;(z') at the film surface; I/;o=I/;(!h) =I/;( -!h) if h is the thickness of the film element. From (3.9) one obtains
I/;o=if!-'Po, (3.14)
where if!=4>(!h) =4>( -!h) is the value of the potential 4> at the surface. As the charge distribution in all elements is now fully known if! may in principle be calculated from (3.4). Since if! is assumed constant on each element not all details of the charge distribution are relevant, however, and in fact only the total charge of each element is required. As an example, for a plane circular film of radius R, thickness h«R with a constant charge q per unit area of film one calculates for the potential if! in the center element of the disk,
if!=q(27rR-7rh) , (3.15)
I/;(r) =4>(r) -'Po
in such a way that
apart from terms which tend to zero as R increases. In (3.9) an element at a distance r from the center the potential
if! is lower; to lowest order in (r/R) the potential decreases quadratically.I6
(3.10)
If no is defined by
J.lo=kT logno, (3.11)
the number densities of cations and anions are given by
n±(r) =no exp[=:t=et/;(r)/kTJ (3.12)
11 Y. Ikeda, J. Phys. Soc. Japan 8,49 (1953). 12 G. M. Bell and S. Levine, Trans. Faraday Soc. 53, 143 (1957). 13 H. Margenau and G. M. Murphy, The Mathematics of Physics
and Chemistry (D. Van Nostrand Co., Inc., New York, 1943), Vol. 1, p. 195.
IV. VARIATION OF DOUBLE-LAYER FREE ENERGY WITH PARTICLE NUMBERS
As the contribution of a flat element to the free energy depends on a few parameters only it is possible to derive thermodynamic relations in which only these parameters enter as variablesP In this section we con-
14 G. Gouy, J. Phys. 9, 457 (1910); Ann. Physik 7, 129 (1917). 15 D. L. Chapman, Phil. Mag. 25, 475 (1913). 16 A. G. Webster, Partial Differential Equations of Mathematual
Physus (Dover Publications Inc., New York, 1955), p. 346. 17 See also B. V. Deryaguin, G. A. Martynov, and Yu. V.
Gutop, in Research in Surface Forces, B. V. Deryaguin, Ed. (Consultants Bureau Enterprises, Inc., New York, 1966), Vol. 2, p.9.
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FLUCTUATIONS OF FREE LIQUID FILMS 1181
sider variations in free energy for constant area A and thickness h of the film element. If we now also allow the surface charge density to vary we obtain for the variation of ffd from (3.3), (3.4), (3.6), and (3.7),
15ffd= 1'0 [p,+15n+(r) +/J-15n_(r) Jd3r+ Is cp(s)15u( s) d2s.
( 4.1)
Hence the contribution to the change in free energy by a single element is
(4.2)
where N +~ and N _~ are the numbers of cations and anions in the bulk of the element and where we have defined the total surface charge of the element as an integral over its surface part S,
(4.3)
We also define
process no or JJ.o=kT logno is regarded as a constant. The appropriate function to consider is therefore E~(JJ.o, Q~) for which
E.(JJ.o, Q.) =- r'l/;o(JJ.o, Qv')dQv'+E.(JJ.o, 0). (4.9) o
The constant of integration E~(JJ.o, 0) is easily determined from (3.3). ForQ=Q.=O the number densities of cations and anions are constant and equal to no and one is left with an ideal-gas contribution,
E.(JJ.o, 0) =-2nokTV=-poV, ( 4.10)
where V =Ah is the volume of the film element and po is the osmotic pressure of the neutral system. Verwey and Overbeek called the integral in (4.9) the electric part of the free energy18 j the precise relation with their expression is
E~(JJ.o, Q~) =2AF.(no, u= -Q./2A) -poV. (4.11)
Their free energy F, which we shall denote by Fyo, is directly related to the function G.(JJ.o, 1/;0), viz.,
G~(JJ.o, 1/;0) =2AFyo (no, 1/;0) -poV. (4.12) Q.=e(N+.-N_.) ,
Q=Q.+Qv, Hence the functions E., F~, and G~ are explicitly known (4.4) from the solution of the PB equation in plane geometry.
With the aid of (3.10), (3.14), and these definitions, we may write
(4.5)
The last term is the contribution to the change in outside field energy defined in connection with (2.2), whereas the first two terms arise from the change in free energy of the element proper. This part we shall denote by F.,
(4.6)
The free energy F. is a function of the variables N~ and Q., i.e., of the numbers of ions in the bulk of the element. It may be convenient to transform to other variables by means of a Legendre transformation. We define the following thermodynamic functions:
E.=F.-N.JJ.o,
G. =F.-N.JJ.o+Q.l/;o,
with the thermodynamic relations
dE. = -N.dJJ.o-l/;odQ.,
dG. = - N .dJJ.o+Q.dl/;o.
(4.7)
(4.8)
Each of these functions must be regarded as a function of the variables that are varied in (4.8).
It is now easy to derive the relation with the doublelayer free energy as defined by Verwey and Overbeek.4
They considered films for which the total charge Q vanishes so that Q. = -Q •. Moreover, in their charging
V. EQUILffiRIUM WITH THE SURFACE PHASE
It will be assumed that the equation of state of the two-dimensional surface phase is known and hence that the surface free energy of an element is a known function of the numbers of adsorbed ions N +. and N -I, or alternatively of N.=N+.+N_. and Q.=e(N+.-N-a)' The variation in surface free energy of an element when the number of adsorbed ions is altered is given by
dF. = JJ.+.dN +.+ JJ._.dN_.
= JJ..dN.+cp.dQ., (5.1) where
JJ.-.=JJ..-ecp •. (5.2)
If there is a free exchange of particles between surface and volume the numbers of adsorbed ions adjust themselves in such a way that the total free energy F=F.+ F. for constant N=N~+N. and constant Q=Q.+Q. is minimal. From (4.6) and (5.1) it follows that in equilibrium,
JJ.o =JJ.. = (aF./aN.)Q"
1/;0= -<{J.= - (f)F./iJQ.)N.' (5.3)
Since JJ.o and 1/;0 must here be regarded as functions of N+v, N-v determined by the explicit expression for F. the two equations (5.3) may in principle be solved for N. and Q •. This determines the adsorption of ions for
18 Reference 4, p. 62.
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1182 B. U. FELDERHOF
given total numbers N + and N _ or for total number of ion pairs N and total charge Q in the element. Thus the total free energy F becomes a function of Nand Q only with the thermodynamic relation
dF=d(Fv+F.) =/J.odN -if;odQ, (504)
where /J.o and if;o as functions of Nand Q are determined by (5.3).
It may again be advantageous to transform to other variables and define
E.=F.-N./J..,
G.=F.-N./J..-Q.«!.,
with the thermodynamic relations
(5.5)
and using (4.7) and (4.12)
'Y=Gv/ A = 2Fvo(/J.o, if;o) -Poh. (604)
The pressure p. is easily determined by considering variations in thickness at constant area and potentials /J.o and if;o. Hence one finds19
P.= -2 (aFvo/ah) I'O,~o+Po
= 2nokT cosh[eif;(O) /kT], (6.5)
where if;(O) is the potential if; at the midplane. From (6.1) and (6.3) one finds a simple equation for "I,
(6.6)
where n.=N./A and q.=Q./A are the number of ions and charge in the bulk per unit film area.
dE. = -N.d/J..+«!.dQ.,
dG.= -N.d/J..-Q.d«! •. The variation in surface free energy is now given by
(5.6) the thermodynamic relation
Similarly one may define E(/J.o, Q) =E.+E. and G(/J.o, if;o) =Gv+G. with thermodynamic relations that follow from (504).
If there is a free exchange of ions between different elements of the film the number of ions N and charge Q of each element adjust in such a way that the total ion free energy
5'ion= L(F.,i+F.,i+!<I>jQi) (5.7) j
is minimal. From (4.5) and (504) it follows that this requires /J.o and «!o= <I>-if;o to be constant along the film. Since <I> is the boundary value of the potential cf> which outside the film volume satisfies t.cf>=O this is essentially a mixed boundary condition for the potential equation.
VI. VARIATIONS OF AREA AND THICKNESS
When variations in area A and thickness h of a flat element are taken into account it is again convenient to consider the double-layer system and surface phase separately. The free energy F. may be regarded as a function of A and h or alternatively of A and V =Ah. We define tensions a and "I and the pressure p. by the thermodynamic relations
dF.=/J.odN.-if;odQ.+adA-p.dV
=/J.odN.-if;odQ.+'YdA -P.Adh, (6.1)
with the obvious relation
(6.2)
These quantities are known explicitly from the relation (4.12) with the VO free energy. A Gibbs-Duhem relation may be established by noting that F.(Nv, Q., A, h) is a homogeneous function of the first degree in the variables N., Qv, and A. Hence it follows that
(6.3)
dF8 =/J..dN.+«!.dQ.-2ITdA, (6.7)
where 2IT is the sum of the spreading pressures of upper and lower surface separately. As F. is a homogeneous function of the first degree of N., Q., and A one has
F.=N./J..+Q.«!.-2ITA (6.8)
or G. = - 2ITA. From (6.7) and (6.8) follows the Gibbs adsorption equation
(6.9)
where n 8 =N./ A and q.=Q./ A =2CT are the number of surface ions and the surface charge per unit film area.
When there is a free exchange of ions between surface and bulk the chemical potentials must be equal: /J.o= /J.. and -if;o=«! •• In this case
(6.10)
is a function of the variables /J.O, if;o, and h, and
dg= -nd/J.o+qdif;o-pvdh, (6.11)
where n=n.+n. and q=q.+q. as functions of /J.O, if;o, and h follow from the equilibrium conditions (5.3). The pressure P. as a function of these variables is again given by (6.5).
VII. FLUCTUATIONS ABOUT PLANE EQUILmRIUM
A liquid film in thermal equilibrium which, on the average, is planar and neutral will exhibit small thermal fluctuations in shape. The average square amplitude of these fluctuations may be calculated from the wellknown Einstein theory20 with the aid of the thermodynamics of the preceding sections. The small corrugations of the surface may be observed by light scattering and one task of the theory is to predict the intensity and angular dependence of the scattered light.
19 Reference 4, p. 91. 20 A. Einstein, Ann. Physik 33, 1275 (1910).
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FLUCTUATIONS OF FREE LIQUID FILMS 1183
Since the amplitudes are small the fluctuations in shape may be described by a linear theory and may be resolved as superpositions of plane wave modes. Although in this paper we do not make a detailed study of dynamics it is necessary to specify these modes in some detail. We choose coordinates such that z=O is the midplane of the film in complete equilibrium. Then there are two independent modes for any given twodimensional wave vector k = (kx, ky), viz., a squeezing mode in which upper and lower film surface move out of phase, and a stretching mode in which upper and lower surface move in phase. For a given wave vector k the frequencies of the two modes differ but both frequencies increase monotonically as the wavelength becomes shorter.
The frequency of the waves should be compared with the relaxation times of the ions. The time necessary for an electric double layer to build up in the bulk2! is of the order 10--7 sec so that for frequencies smaller than 107 sec! we may assume that the double-layer system at any time adjusts itself to the shape of the film. On the other hand the redistribution of ions within the surface phase and the establishment of equilibrium between surface and bulk is a relatively slow process which takes a relaxation time r of the order of milliseconds or longer.2! Hence one must distinguish between slow wave modes with frequency smaller than 1/r and fast modes with frequency larger than 1/r. The average square amplitudes of the modes differ in these two frequency regimes.
In the equilibrium situation the film is planar with constant thickness ho, and neutral, so that the total charge of each film element is zero and the surface potential <I> vanishes identically. The ion distribution is characterized by potentials fJ,0 and </'0= -1/;0 where 1/;0 is determined by the equilibrium between bulk and surface phase. The potential fJ,o=kT logno is determined by the total number of ions or, if the film is in open contact with a neutral reservoir, by the ion density no in the reservoir.
Let us now consider a small distortion in the shape of the film. Due to the rapid exchange of ions between the bulk parts of different film elements the potentials fJ,0 and CPo = <I>-1/;o are constant throughout the film. The values may be taken to be those of the equilibrium situation, either because of the equilibrium with a neutral reservoir or because a fluctuation in a small part of a large film is considered. In the slow regime there is time for redistribution of ions between surface and bulk; the surface potentials fJ,. and cps are equal to fJ,0 and -1/;0, respectively, and 1/;0= <I>-cpo must be such that the electrostatic potential <I> is consistent with the total charge Qj of each element. In the fast regime the ions adsorbed on the surface remain adsorbed in the corresponding surface element of the distorted film. The volume charge is determined by the local value of
21 J. T. Davies and E. K. Rideal, Interfacial Phenomena (Academic Press Inc., New York, 1961).
1/;0 = if!-cpo which again adjusts itself in such a way that if! is consistent with the total charge distribution.
VIII. FLUCTUATIONS IN THE SLOW REGIME
In order to establish the probability distribution for fluctuations about equilibrium one must, according to Einstein's theory, evaluate the appropriate thermodynamic function to second order in the deviations. In the slow regime there is a free exchange of ions between all elements and fJ,0 and cpo are constant throughout the film. The appropriate thermodynamic function is therefore
g=5'-;nfJ,O-Qcpo= L(Gj-!if!jQj) , (8.1) j
where ;n is the total number of ions, Q is the total charge, and the sum runs over the flat elements of the film. The value of this function for the distorted film should be compared with its equilibrium value.
It is more convenient to change notation and replace the sum by an integral
g= fg(a)d2a-~fif!(a)q(a)d2a, (8.2)
where g and q are the free energy and charge per unit film area defined in (6.10) et seq., and where a runs over the midplane of the distorted film. This is well defined in the present approximation where the film is regarded as an assembly of flat elements. The free energy per unit area g(fJ,o, 1/;0, h) is a function of a through its dependence on the local values of 1/;o(a) and h(a). To second order in the deviations from the equilibrium values OIPo(a) and oh(a), defined by
h(a) =ho+oh(a) , (8.3)
one obtains, using the fact that in equilibrium the charge density q and the potential <I> vanish identically,
J[ ag 1 a2g a2g g= go+ ah oh(a)+ 2 at/;02 OIPo(a) 2+ at/;oah OIPoOh
(8.4)
where go=g(/-Io, -CPo, ho) and all derivatives have their equilibrium values. Rewriting the last term by use of
oif!(a) = o1/;o (a) ,
oq(a) = (aqla1/;o)o1/;o(a) + (aqlah)oh(a) , (8.S)
and employing (6.11) one finds
1f{~ ~} + - -g- o<l>(a)oh(a)+ -h2 [oh(a)]2 d2a. (8.6) 2 a1/;oah a
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1184 B. U. FELDERHOF
Since the characteristic variation length of the distortions we consider is large compared with the thickness of the film, the charge oq(a) may be taken as located at the surface a. This surface charge density then determines the jump in the normal derivative of the potential cp across the film and from (8.5) one finds the mixed boundary condition
-4I1In· Vo4>] (a) = (aq/iNoWP(a) + (aq/ah)oh(a) ,
(8.7)
which must be met in solving the potential equation il4>=O in the space outside the film. Thus one finds in princi pIe o~ ( a) as a function depending on the thickness fluctuations and the position of the midplane of the distorted film.
The distortions of the film are most conveniently expressed in terms of the elevations r 1 (x, y) and r 1 (x, y) of the upper and lower surface. Accordingly we define the surface a by
(8.8)
where ao= (x, y, 0) is the midplane of the equilibrium film and e. is the unit vector in the z direction. By this definition o~ and oh become functions of ao. To second order in the elevations one has
oh(ao) =rt-r I-lho[a(rt+r 1)/aao]2,
d2a= p+l[a(rt+r 1)/aao]2}d2ao. (8.9)
Hence if 9 is desired to second order in the elevations, o~ (ao) need be calculated only to first order. To this order the boundary condition (8.7) becomes simply
-411'[aocp/az](ao) = (aq;aif;o)o~(ao) + (aq/ah) (rt-r I),
(8.10)
where the jump [ ] is to be evaluated at the equilibrium plane ao. The potential equation is easily solved with this boundary condition yielding a relation between
(8.11)
and (rt-r I) k similarly defined, viz.,
(8.12)
In the long-wavelength limit 1 k 1---+0 this amounts to
(8.13)
In this limit the function 9 in (8.6) for the distorted
film becomes to second order in the elevations,
+f{~ (a-2IT) (a(rt+r 1))2 _ ~ [(apv) (aif;o) 8 (lao 2 aif;o /0 ah q
+ (:)J (r,-r 1)2} d2ao, (8.14)
where use has been made of (6.2), (6.10), and (6.11). The first term in (8.14) is the equilibrium value of g. The second term vanishes identically since the total film volume does not change in the incompressible fluid motion. The third term gives the contribution of the ions to the resistance against stretching and the last term gives the ion contribution to the resistance against squeezing. We shall discuss these contributions in Sec.X.
IX. FLUCTUATIONS IN THE FAST REGIME
As mentioned in Sec. VII, for sufficiently high frequencies there is no time for exchange of ions between surface and bulk. Ions adsorbed at the surface remain adsorbed in the corresponding surface element of the distorted film. But there is a rapid exchange of ions between the bulk parts of different film elements so that p.o and CPo are kept constant throughout the film. The appropriate thermodynamic function to use in the study of fluctuations is therefore
9.= 5'-;r[.p.o-c;).cpo = L)F.,j+G.,j-!~jQj+~jQ •. j), j
(9.1)
or changing notation to a continuum description
g.= f [f.(a) +g.(a) -!~(a)q(a) +~(a)q.(a) ]d2a,
(9.2)
where a is defined such that the correspondence between an element of area d2a of the distorted film and the element d2ao of the equilibrium film is given by the actual motion. The fact that the surface ions remain adsorbed is expressed by
where q.o and n.o are the equilibrium surface densities. As will be shown by a more detailed analysis in a
forthcoming paper, under realistic assumptions about the fluid flow the surface a may be taken to be identical with (8.8). This enables us to expand (9.2) to second order in the elevations like in the previous section. The first term in (9.2) may be expanded to first order in
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FLUCTUATIONS OF FREE LIQUID FILMS 1185
the particle densities as
where according to (9.3) the deviations 5n. and 5q. are given by
[q.o+5q.( a) ]d2a= q.od2aO,
[n.o+5n.(a) ]d2a=n.od2aO. (9.5)
Hence using (6.7) and (6.8) one finds
f j.(a)d2a= f (j.o+2IT)d2ao-f 2ITd2a. (9.6)
The remaining terms in (9.2) may be expanded in the same way as in the previous section. To second order in the elevations one obtains
- ~ [(ap.) (~o) + (apv
) J(tt-t IP} d2ao. (9.7) 2 ~o h ah q. ah y,o
The first term is the equilibrium value of 9. and the second term again vanishes identically. Only the last term is different from (8.14) in that (~o/ah)q has been replaced by (at/to/ah)q,.
X. CONCLUSIONS
We have determined the thermodynamic functions relevant to the iluctuations in shape to second order in the elevations of the upper and lower film surface from the equilibrium planes. The results (8.14) and (9.7), which may be directly transformed into intensities for light scattering,6.22 apply in two different regimes, namely the slow and the fast regime corresponding to frequencies of the surface waves smaller or larger than l/T, where T is the characteristic relaxation time for adsorption of ions at the film surface. It turns out that in both regimes the resistance against stretching is the same and is given by a-2IT, where a is the
211 F. P. Buff, R. A. Lovett, and F. H. Stillinger, Phys. Rev. Letters 15, 621 (1965).
double-layer tension of the equilibrium film which according to (6.2), (6.4), and (6.5) equals
a=2[Fvo (J.!o, t/to) -ho(iJFvo/iJh)l'o,y,o]. (10.1)
IT is the spreading pressure on the film surface in equilibrium.
The resistance against squeezing differs in the slow and the fast regime, but in both cases it may be written as - (dp./ dh). This is essentially the same expression as given by Vrij6 who, however, did not specify which quantities are to be held constant in taking the derivative. It follows from our analysis that in the slow regime, i.e., for long wavelengths, the resistance is given by [cf. (8.14)]
_ (apv) (at/to) _ (apv) = _ (dPv) . (10.2) ~o h ah q ah 1/10 ah ~
As follows from (9.7) in the fast regime the resistance is given by -dpv/ dh with qv being held constant.
In the slow regime the resistance is proportional to the variation of disjoining pressure with thickness while the film is kept neutral. To determine this derivative it is essential to take the surface phase into account. An often used approximation is to set t/to=X constant. According to (5.3) this is equivalent to
F.(N., Q.) =F.(N" 0) -XQ •. (10.3)
More generally (at/to/ah)" is determined by (5.3) and the surface equation of state.
In the fast regime the resistance against squeezing is given by - (apv/ah)q, as determined by (6.5) and depends only on the ion distribution in the bulk. It may therefore be calculated explicitly independent of the surface equation of state.
In deriving (8.14) and (9.7) use has been made of a long-wavelength limit. This limit is consistent with the assumption that the film may be regarded as an assembly of ilat elements. It is clear that correction terms proportional to [iJ(tt-t j)/aao]2 cannot be obtained correctly on the basis of this assumption. In order to obtain correction terms to (8.14) and (9.7) a more detailed analysis is required and this will be the subject of future work.
ACKNOWLEDGMENT
The author wishes to thank Dr. A. Vrij for suggesting this problem and for many stimulating discussions.
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