3024
E. R. Cooper and J. A. Mann
A Kinetic Theory Model for Insoluble Monolayer
roperties. The Dilute Gas Case’
. R . Cooper and J. Wdin Man& Departments of Chemistry and Biochemistry and Biophysics. Unwersity of Hawaii, Honolulu, Hawaii 96822 (Received December 4, 1972) Publication costs assisted by The Petroleum Research Fund
A microscopic description of momentum transport in gaseous insoluble monolayers is developed. The substrate is taken into account by means of a Fokker-Planck or friction term in the kinetic equations. ru’umerical estimates are obtained for the shear and dilational viscosity coefficients by employing rigid ellipsoids to model the molecular interaction potential. Introduction The object of this paper is the systematic presentation of a microscopic model for calculating surface transport coefficients for the special case of an insoluble gaseous monolayer spread at the air-water interface. Calculations are reported only for the dilute gaseous monolayer case. As an obvious consequence of the limitations on obtaining numerical estimations from accurate microscopic theories, calculations of surface transport coefficients are limited by the necessity of adopting simplifying approximations. Therefore it is pertinent to comment on the reasons for attempting what must be by necessity an imperfect computation. It is well documented that transport properties can be associated with a phase interface.2a More specifically, the “surface viscosity” of monomolecular films spread at the air-water interface has been studied experimentally by numerous authors.2b That a viscosity coefficient can be associated with monomolecular films is convincingly demonstrated by observing two-dimensional flow patterns in terms of experimental arrangements adapted from the standard methods of determining three-dimensional viscosity coefficients.2 Unfortunately there is always the question of including accurately the effect of the substrate on the two-dimensional measurement. For example, one experimental arrangement involves observing the time damping of an oscillating disk carefully placed in the surface region. Indeed, the oscillations of the disk do damp out more quickly in the presence of a monomolecular film. However, there is a major difficulty in subtracting out the substrate effect on the disk in order to calculate a surface viscosity rigorously and accurately. The hydrodynamic problem is discussed by Goodrich and Chatterjee.3 As a consequence of such difficulties, the comparison of experimental and theoretical results involving surface viscosities can be misleading since there must be doubts that the experimentally determined numbers represent the “abstract” surface viscosity. A simple calculation of the surface viscosity coefficient by dimensional arguments illustrates a difficulty in comparing even orders of magnitude of the theoretical and observed surface viscosity coefficients. Suppose that the interfacial region is between 1 and 10 nm thick and involves a fluid monolayer. Then one would expect the surface visg/sec, corresponding to a cosity to be roughly 5 X fluid viscosity of 0.05 P. The surface viscosity coefficients of a gaseous monolayer should be even smaller. In contrast the surface viscosity of such monomolecular films is reThe Journal of Physical Chemistry, Vol. 77, No. 25, 1973
ported to be of the order of 1 P 4 g/sec or greater.*b A more careful calculation was reported by Blank and Britten4 who used a fluctuation theory approach for the calculation of surface transport coefficients. Their calculation of the surface viscosity coefficient was still many orders of magnitude too small. The viscosity coefficient estimated by the simple dimensional argument and that calculated by Blank and Britten4 are in reasonable agreement but such numbers do not agree at all with surface viscosity coefficients reported by the experimentalists.2b As Blank and Britten4 point out, a possible resolution of this interesting dilemma is that the experimental techniques are not reporting the surface viscosity coefficients as defined in the theoretical models. However, the Blank and Britten approach probably underestimates the substrate contribution to surface transport.2 An important question, then, is whether a more detailed microscopic model of two-dimensional transport could result in a satisfactory comparison with experimental results. Since kinetic theory has been successfully developed for dilute g a ~ e s ,we ~ , feel ~ that we can obtain reliable estimates of both the shear and dilational viscosity coefficients in the dilute region corresponding to a surface area/molecule greater than about 1000 A2/molecule. Further, we hope that our approach will shed some light on the difficulties encountered in understanding the experimental results in the dense regions and hopefully provide a framework for future considerations of such regions in terms of kinetic theory. Although we realize that the surface film is a threedimensional, binary fluid mixture, we model it as a twodimensional single-component fluid, where the interaction between the substrate and monolayer is treated as a friction or Fokker-Planck term in the equation for the singlet distribution function. Therefore, we ignore mass transport into and out of the two-dimensional layer and lump the complex monolayer-substrate interaction into a friction term. Practical advantages are gained by such a description. For the dilute monolayer gas, we can solve the singlet equation and obtain numerical estimates of the transport coefficients based on a rigid ellipsoid interaction model. Further, the extension to denser gases (at least for a spherical interaction) can be expressed quite naturally in terms of the Rice-Allnatt7.8 kinetic theory for liquids. Since their model also employs friction constants (molecular friction constants), we can add the substrate friction constant to the soft-potential friction constant in the Rice-Allnatt theory.
3025
Kinetic Theory Model for Transport Properties
We show that the substrate-monolayer interaction term in the singlet equation can be cast into a Fokker-Planck term for rigid-sphere interaction potentials. We assume that this is also the case if the monolayer molecules are treated as rigid ellipsoids, and that a term exists for both translational and rotational motion. We thus obtain a singlet equation for a gas of rigid ellipsoids which are constrained to move in two dimensions with the linear and angular momentum vectors perpendicular a t all times. We first discuss the singlet equation and its solution and then present numerical estimates of the surface shear and surface dilational viscosity coefficients for the ellipsoid model.
Theoretical I. Singlet Equation. In the absence of external fields, the equationg determining the singlet distribution function f l (rl,vI,M1,t) for a dilute gas of rigid ellipsoidal bodies confined to a plane is
where v1 and rl are the velocity and location of the mass center of molecule one in the surface, MI is the angular momentum (always perpendicular to u1 and therefore in the direction of the surface normal), and for rigid ellipsoids
where [ is the substrate friction coefficient for the translational motion and [’ is the friction coefficient for the rotational motion. (Note: we have assumed velocity independent friction coefficients.) Here us is the substrate stream velocity, m is the molecular mass of the monolayer, 1 is the moment of inertia of the monolayer, k is Boltzmann’s contant, and T is the temperature. The well-documented coupling of the substrate to monolayer is taken into account explicitly and simply by the inclusion of friction coefficients in eq 6. The basis for our assertion of the friction terms in eq 6 is that we can show that such is the case for a binary mixture of rigid spheres with the mass of one component much larger than the mass of the other (see Appendix A). We note that the mass ratio of molecules that form insoluble monolayers and water is at least 10:1. 11. Linearization of the Singlet Equation. T o generate the “normal” solutions to the singlet equation, we introduce an ordering parameter t and write f1 = f;’O’ + ~ f i l ~ l + . . . . This parameter enters the singlet equation as
3c(f 1, f2) =
+
(2~)-~fddl.fddJdvJd M JdE S,“’& g{fl’fi’ i.g
>0
flf2l
E.g
(2)
s ~ , , , ' " ( ~ ~B@ ) [ w ] ~ ~(261 ~0* 9 c t
(16)
The hydrodynamic equations, which give the time derivatives of n, u, and T ( i e . , of f ' o ' ) , can be constructed by multiplying eq 9 by the summational invariants $i and integrating. Thus from
BO00 =
B O I O Ago01 +
=o
(28)
Also, since there can be no coupling through J of tensors of different rank, we have that q = 0,2. For our expansion of B, the pressure tensor becomes
p = pU(2) - kT[BZ@@ - U(2)B@"]@E?!! ar
(29)
where Woois traceless and symmetric on its first two indices. Since the two-dimensional space is assumed isotropic, there exist coefficients b200 and bolo such that 200
we obtain the zeroth order (in e ) hydrodynamic equations
(18)
(27 )
0
and
-
b200
Ja)(2)
(30)
and (3 1)
Bo10 = bolOu(2)
where ( B ~ ( ~ ' ( 2 ) ) = ~ ~1/2(6,161~ kl becomes
+ 6,izS11
- S r 1 6 k ~ ) .Thus P
and
(20) wherepl = nlkT1. We now can rewrite (9) as
where [W](n).is a rank n irreducible Cartesian tensor,12 O n denotes a contraction on n indices and (U@')ij= 6 i j for ij = 1,2.(For convenience we have now dropped the subscript one.) We now approximate 4 by
Since there can be no visco--thermal coupling in the absence of external fields, eq 21 separates into the two equations
where the surface shear viscosity is ;= Y2hTbzo0,the surface dilational viscosity is K = zk7'b010, and S = %[(du/dr) (du/dr)t] - l/zUJ) (d/ar)-u. Here (du/arIt is the transpose of iiular. We now truncate the expansion for B after the inclusion of a few lead terms. We thus approximate B by B = $'Bl +Q2B2 4 Q302B3 -k Q402B4 (33)
+
+
where = So'l'(WZ) - 2s-1 2 l ) ( Q 2a2 ) , = [W] 2 ) , a3 = S _ ~ , Z ' ~ ' ( R ~ ) [ Wa* ] ' ~=' ,Sz'1'(W)[W](2),B1 = BO'", B2 = Bzoo,B3 = BZo1,and B4 = Multiplying eq 23 by each of the and integrating6 we obtain equations for the BZ and hence 11 and K . We find that
o1
l 7 = hT/3,, and r] =
kT
(34)
- - J212/Jqq -
~ { J 2 2
(J23 -
- J~~'//JM (35)) J - ~
J24J34/Jd2/(-'.Tx
where
3,, = J~~-Ij-/mn -I- 211/rnn To solve for 4 then, we must determine A and B, from which we obtain the thermal conductivity and viscosity, respectively. Even though we are concerned only with a solution of the viscosity equation (23), we note that the solution technique for the thermal conductivity equation (24) is quite similar. III. Solution of the Viscosity Equation. The contribution (to first order in the gradients) to the pressure tensor from the monolayer is given b y
For a three-dimensional gas B is usually expanded in products of irreducible tensors in W and R, but since W-R = 0, we only need to expand B as The Journal of Physical Chemistry, Vol. 77, No. 25, 1973
Jd3= J~~ + l / m n + t'/rnn c?,,
=
J
+ 0 = depths of the interfacial region = surface tension
= pressure tensor of the substrate in the bulk = self-diffusion coefficient
6(A,)(3) + 13.125(A,)(~) - 15(A,)(7) and
References and Notes
[Qz WOW; WoWQ2]= 3(1)(') - (1)(3)+ 8(A1)(') +
(1) Acknowledgment is made to the Donors of The Petroleum Research Fund, administered by the American Chemical Society, for the partial support of this research. (2) (a) G. L. Gaines, "Insoluble Monolayers at Liquid-Gas Interfaces," Wiley-lnterscience, New York, N. Y.. 1966: (b) M. Joly in "Surface and Colloid Science," Vol. 5, E. Matljevic, Ed., Wiley, New York, N. Y., 1972. (3) F. C, Goodrich and A. K. Chatterjee, J . Colloid interface Sci., 34, 36 (1970). (4) M. Blank and J . Britten, J . ColloidSc!., 20, 789 (1965) (5) S. Chapman and T. G. Cowling, "The Mathematical Theory of NonUniform Gases," Cambridge University Press, Cambridge, 1952. (6) J. 0. Hirschfelder, C. F. Curtiss. and R. 8 . Bird, "Molecular Theory of Gases and Liquid," Wiley, New York, N. Y., 1964. (7) S . A. Riceand A. R. Allnatt, J. Chem. Phys., 34,2144 (1961). (8) S. A. Rice and A. R . Allnatt, J . Chem. Phys., 34,2156 (1961). (9) D. K. Hoffman and J. S. Dahler, J . Stat. Phys., 1, 521 (1969). (10) E. R. Cooper and D. K. Hoffman, J. Chem. Phys., 53, 1100 (1970). (11) Equation 9 Is not a Fredholm integral equation of the second kind
2(A1)(3)+ 4.7?1(A,)(~) -
+ 27(A,2)(3)+
27(A12)(') - 15(A12)(7) - 15(A1Az)(7) Here (A1)(V) = (AzI'), WOW = [W](2) z= WW - 1/2WU'2),and the semicolon means that we have corrected all tensors, beginning on each side of the semicolon and contracting nearest indices.
Appendix C fl(r, v , M, t) = singlet distribution function for ellipsoids r = location of molecular mass center V = velocity of molecular mass center The Journal of Physical Chemistry, Vo!. 77, No. 25, 1973
3033
Helical Conformations of a Tetrapeptide of Poly-L-alanine with a symmetric kernel. J. A. R. Coope, R. F. Snider, and F. R. McCourt, J. Chem. Phys., 43,2269 (1965). D. K. Hoffman, J. Chem. Phys., 50, 4823 (1969). E. Boyd and W. D. Harkins, J. Amer. Chem. Soc., 61, 1186 (1939). M. J. Povich, Thesis, Universityof Hawaii, 1972. R. J. Mannheimer and R. S. Schechter, J. Colloid lnterface Sci., 27,342 (1968); 32, 195 (1970); 32, 212 (1970). F. C. Goodrich and L. H. Allen, J. Colloid lnterface Sci.. 37, 68 11971) \ . _ . .,.
N. Lifschutz, M. G. Hegde, and J. C. Slattery. J. Colloid lnterface So., 37,73 (1971). J. A. Mann in "Techniques of Surface Chemistry and Physics," R. J. Good, et ai.. Ed., Marcel Dekker, New York, N. Y., 1972.
(20) R. S. Hansen and J. Ahmad, Progr. Surface Membrane Sci., 4, 1 (1971). (21) . . E. H. Lucassen-Reynders and J. Lucassen, Advan. Colloid lnterface Sci., 2, 347 (1969). (22) M. G. Hegde and J. C. Slattery, J . Colloid lnterface Sci., 35, 183 (23) (24) (25) (26)
11971) \ ' - . ','
C. Huh and L. E. Scriven, J. Colioid lnterface Sci., 35,85 (1971). J. A. Mann and J. Ahmad, J. Colloid lnterface Sci., 29, 158 (1969). J. Ahmad and R. S. Hansen, J. Colloid lnterface Sci., in press. R. J. Bearman and J. G. Kirkwood, J. Chem. Phys., 28, 136 (1958). (27) E. K. Sakata and J. C. Berg, Ind. Eng. Chem., Fundam.. 8, 570 (1969) (28) D. Montgomery, Phys. Fluids, 14, 2088 (1971).
A CNDO/2 Calculation on the Helical Conformations of a Tetrapeptide of Poly-L-alanine. V. The $-$ Energy Surface' Hans Stymne, Gunnar Wettermark, Division of Physical Chemistry, The Royal Institute of Technology, Stockholm 70, Sweden
Robert Schor," Department of Physics and lnstitute of Materials Science, The University of Connecticut, Storrs, Connecticut 06268
and Carl W. David Department of Chemistry, The University of Connecticut, Storrs, Conneciicut 06268 (Received July 79, 1973) Publication costs assisted by The University of Connecticut Research Foundation
The ground-state potential energy surface for the helical conformations of a tetrapeptide of poly-L-alanine has been calculated using the CNDO/2 method. The potential energy surface contains four nonequivalent minima. The absolute minimum is found in a region close to both the right-handed a helix and the right-handed 310 helix. A second local minimum of about 2 kcal/mol of residue higher is found near the corresponding left-handed helices. The 2, helix may be represented by the third minimum of about 3 kcal/mol of residue. The last local minimum of about 5 kcal/mol of residue is near the fully extended chain conformation. The results are compared to our previous calculations on the same system as well as to our calculations on the tetrapeptide of polyglycine.
Introduction Theoretical studies of the conformations of isolated helices (under vacxum) of polypeptide chains with intramolecular interactions have been carried out by many workers2x3 using semiempirical potential functions for rotation around single bonds, nonbonded interactions, dipole-dipole interactions between amide groups, and hydrogen bonding potential energy functions. More recently semiempirical quantum mechanical techniques have been used to study glycyl and alanyl r e ~ i d u e s ,polypeptide ~ chains,5 and model peptide molecules.6 A detailed study by extended Hiickel theory of a polypeptide chain of polyL-alanine long enough to incorporate an intramolecular hydrogen bond (see Figure 1) has been presented.? In the present work, vve present the corresponding results using the CNDO/Z method. It should now be possible to make a preliminary but more detailed comparison of the two methods on these systems and of the differences between the tetrapeptide of glycine and poly-L-alanine.
Method Santry's modifications of the CNDO/2 method*-i2 was applied. The CNDO/2 method provides an approximate SCF solution to the LCAO molecular Hartree-Fock equations, in which all valence electrons are included and in which electronic repulsion is explicitly introduced. The methods of determining the coordinates of the atoms in the helical conformations of the polypeptide chain as shown in Figure 1 is due to NBmethy and Scheraga.i3 The peptide unit is considered to have a rigid planar structure with fixed bond angles and bond lengths. The coordinates of the atoms in a peptide unit for the bond angles and distances taken from Leach, NBmethy, and Scheragai4 are given in Table I. Figure 2 shows a representation of a dipeptide segment of poly-L-alanine. The new conventionsi5 for the rotation angles (b and fi are used in the present work. (The rotation angles (b and $ as given by the former conventioni6 are related to the new rotation angles '$ and # by @new = '$old - r, $new = $old - r.1 The Journal of Physical Chemistry, Vol, 77, No. 25, 1973
