Elastic moduli in monolayers

JOURNAL OF COLLOID SCIENCE 13 500--507

(1958)

ELASTIC MODULI IN MONOLAYERS N. W. Tschoegl Bread Research Institute of Australia, North Sydney, New South Wales Received December 28, 1957 ABSTRACT The surface elastic moduli are t h e two-dimensional analogs of the elastic moduli in three dimensions. E x p e r i m e n t a l methods are reviewed briefly, a n d it is shown t h a t F o u r t ' s " e l a s t i c i t y i n d e x " for a n oscillating vane o1" needle is dimensionally incorrect, whereas the formula proposed b y T r a p e z n i k o v for a n oscillating disk or ring is correct a n d can be derived exactly b y regarding t h e film as a n elastic annulus sheared in its own plane. A dimensionally correct formula is proposed for an oscillating needle as well as for a n aperiodic needle type surface torsion p e n d u l u m . The relationships between the four two-dimensional elastic c o n s t a n t s of homogeneous isotropie films are derived a n d are shown to be different from the threedimensional formulas. The maximal value of t h e surface Poisson's r a t i o is u n i t y instead of one half. INTRODUCTION

In a monolayer, being by definition one molecule in thickness, bonding between similar molecules can occur in two dimensions only. It is customary to regard such a system as a two-dimensional one and to define for it twodimensional analogs of the relationships in three dimensions. In this sense the rheology of monolayers may be referred to as twodimensional or surface theology in which, for instance, the elastic moduli have the dimensions of surface traction (force per unit length) instead of traction (force per unit area). Similarly, whereas bulk viscosity has the dimensions ML1-T -I, two-dimensional or surface viscosity has MT-', the c.g.s, unit being termed a "surface poise." As the state of aggregation of the molecules in a monolayer is generally different from that in bulk, three-dimensional coefficients ~,' calculated from surface rheological coefficients ks and the thickness d of the monolayer according to the formal relation ~., =

_X" d

[1]

are not identical with the "true" bulk coefficients k. Thus Fourt and Harkins (1) found the "calculated" bulk viscosity of long-chain alcohols to be of the order of magnitude of the bulk viscosity of waxes and pitches. 50o

ELASTIC MODULI IN MONOLAYERS

501

The Two-Dimensional Elastic Moduli Whereas surface (or interracial) viscosity has received a good deal of attention, the other rheological properties of insoluble and adsorbed monolayers have been comparativeJy neglected. The surface compressional modulus K~ can be obtained from the tangent of the surface pressure-area (F -- A) curve as the reciprocal of the surface coed~cient of compressibility (2) ca c~

~-~ .

[21

Methods for the determination of the surface shear modulus G~ include the rotating disk method (3, 4), the resonance frequency method (5), the aperiodic torsion pendulum (6, 7), and the oscillating torsion pendulum method (8-10). No systematic study has yet been undertaken to compare these methods and the published data are too scanty to allow a comparison of the results. The last-mentioned method was developed by Fourt (8), who derived the contribution of the film to the restoring torque of a surface torsion pendulum from the decrease in period as (1 Cs - 47reI ~-~

1)

Ge

,

[3]

where C/is the increase in the effective torsional constant of the system due to the film, I is the moment of inertia of the pendulum, and T and To are the periods measured with the film and the clean surface, respectively. From Eq. [3] Fourt defined an "elasticity index" E ~ - 4~r~I ~ - ~2

~

[4]

for an oscillating vane (or needle) instrument by introducing the third power of the length of the vane 1 as an empirical factor in order to render comparable the values obtained with vanes of different lengths. Fourt's index does not have the dimensions of a surface shear modulus. Trapeznikov (9), working with an apparatus of circular symmetry, used the formula E~

7FI

1)(r 2

~2

~2

2

,

'°'

where rl is the radius of the oscillating disk (or ring) and r2 is the radius (or equivalent radius) of the film. Equation [5] was apparently obtained by multiplying C/with the apparatus constant (6) 1(1 H

=

G

1) 2

"

[61

502

TSCHOEGL

Trapeznikov's formula is dimensionally correct and it can be shown that the elasticity index E~ determined from Eq. [5] is identical with the shear modulus G~. The film contributing to the restoring torque of the surface torsion pendulum m a y be regarded as an elastic annulus sheared in its own plane. The torsional constant Cs of this annulus is subject to the relation

[71

c j o = L,

where 0 is the angle of twist and L is the corresponding torque which must be constant throughout the annulus. It remains then to find an expression linear in 0 and L, and containing G~. Now for small strains G. - s.

[8]

g

where s is the shearing stress and e is the shear strain. From known formula L s - 21rr2 .

[9]

Further, as may be seen from Fig. 1, du

r dO

e = d-~ - -d~-'

[10]

where du is the elemental tangential shift and dO is the elemental angle. The desired relationship in its differential form therefore becomes G, -

L dr 21rra dO"

[11]

Integrating between the limits rl and r2, and 01 = 0, 02 = 0, and using Eqs. [7] and [3] a~ = ~rI ( 1~

~ 1)(713

1) .

Hence, from Eqs. [5] and [12], E~ = G~.

I/0o V FIG. 1. Shear s t r a i n of an elastic a n n u l u s s h e a r e d in its own plane.

[12]

ELABTIC MODULI IN MONOLAYERS

503

The oscillating needle offers certain experimental advantages over the disk or ring. Unfortunately no really satisfactory expression has as yet been proposed for the calculation of the coefficient of surface viscosity or the surface shear modulus determined with an oscillating needle or vane, since the problem is indeed very difficult to treat hydrodynamically. Fourt (8), however, derived a surface viscosity equation from the assumption that the resistance of the film may be localized as rupture at the ends of an oscillating vane and showed that the surface viscosities of long-chain fatty alcohols determined in this way with vanes of different lengths fell within the range of the results obtained with rings and disks of different diameters (1). The "apparatus constant" in Fourt's equation is g -

2

[13]

and combining Eqs. [13] and [3], one obtains the dimensionally correct relation Gs-

8rr2I ( 1 ~ - ~-~

1) Tax

[14]

for an oscillating vane or needle. Applying the same type of reasoning to Langmuir and Schaefer's (6) equation for the surface shear modulus determined by a disk type aperiodic surface pendulum, one finds, for a vane or needle, G~ =

~-

1 ,

[15]

where C is the torsional constant of the pendulum and OH and 0~- are the angular displacements of the torsion head and the needle, respectively. An endeavor to determine a surface Young's modulus Ys was made by van Wazer (5) on adsorbed films of saponin but no attempt to measure the surface Poisson's ratio, it, has been published. Since the four material constants are not independent of each other, Y~ and t*s can be calculated from Ks and G~. This necessitates a consideration of the appropriate relationships which will here be shown to differ from the well-known three-dimensional formulas.

The Relationships between the Two-Dimensional Elastic Constants Van Wazer (5) comparing his surface shear modulus and surface Young's modulus states that elastic theory demands Young's modulus to be about three times as large as the shear modulus. This is true for incompressible bodies in three dimensions, but it will be shown that the surface Young's modulus of incompressible films must be four times as large as the surface

504

TSCHOEGL

shear modulus. Also, while the maximal value of Poisson's ratio is 0.5 in three dimensions, this value is 1.0 in two dimensions. In order to derive the relationships between the four elastic constants in two dimensions we shall consider the case of simple push on a square area of unit length of the film. The resolution of the surface tractions is shown in Fig. 2. The isotropic surface pressure of --1/~F on the four sides of the square cause a uniform compression of --1/~F/K~. The area of the square after compression is 1 - F/2Ks and the length of each side, taking the square root, expanding by the binomial theorem, and neglecting all powers of F/2K~ higher than the first, is 1 - F/4K,. Hence, the isotropic surface pressure on each side of the square causes a contraction of - F/4K, in every direction. The set of deviatoric (shearing) surface tractions will cause a contraction of -1/~F/2G, in the direction of the push and an elongation of 1/~F/2G, at right angles to it [11]. Hence, the resultant total contraction in the direction of the push is F F [16] --~' = - - 4 K ~ - - 4G-~ and the resultant total elongation at right angles to it is 1/

=

F 4K~

F 4G,"

- - - - d - - -

[17]

B u t according to Hooke's law, the contraction - 3 , expressed in terms of the surface Young's moduhis Y, is yF.

--3' --

[18]

-F I I -~.F -~F

1'

! !

w

o{

;

<

½F~---

-- -~

'F

-~ }0 ½F

I I t

I

-gF -~F -F Fie. 2. Resolution of simple push into isotropic ( (- - - -*) components.

)) and deviatoric

ELASTIC MODULI IN MONOLAYERS

505

and the elongation 7 p in terms of Y, and the surface Poisson's ratio #, is 7' - #8y ~F .

[19]

We now obtain, from Eqs. [16] and [18], Ya-

4KaG~

K~+G8

[20]

and from Eqs. [17], [19], and [20] K8 -

G~

~-K~+G~"

[21]

F r o m Eqs. [20] and [21] we further find

Y~

K~ - 2(1 -- ~,)'

[22]

G~ -

[23]

Y~ 2(1 + ~ ) "

Equation [21] m a y be written as K,(1 ,

~ ) = G8(1 + #,),

[24]

from which it is clear t h a t Poisson's ratio in two dimensions cannot exceed unity. This can also be shown in a different way. Consider an infinitesimal extension e of the length of a unit area of the film in one direction. If the film is incompressible, there is no change in area and we have (1 + e)(1 -- ~'e) = 1,

[25]

where ~' stands for ~ (m,~.) • F r o m Eq. [25] we find _ ~,e _ #,e2 = 0,

[26]

and, neglecting the second power of e, ~' = ~ ( . . . . ) = 1.

[27]

T h a t the maximal value of the surface Poisson's ratio must in fact be 1.0 instead of 0.5 is easily realized on considering t h a t in two dimensions an extension in one direction is accompanied b y a contraction in only one other direction whilst in three dimensions it is accompanied b y contractions in the other two directions. Inserting t~ = 1 into Eq. [23] we have, for an incompressible film, Y~ = 4G.~ as asserted above.

[28]

506

TSCHOEGL TABLE I

Relationships between the Four Elastic Material Constants Material constant

Three dimensions

Y 3(1 -- 2~)

Two dimensions

K

Shear modulus

Y G - 2(1 -]- ~)

Young's modulus

Y- - 3K + G

Y~ - - Ks + G~

Poisson's ra~io

3K -- 2G ~- - -

K~ - G~ #~ - - -

Maximal value of Poisson~s ratio

~(m~.) = 0.5

#~(m~,.) = 1.0

Young's modulus (for K = ~)

Y = 3G

Y~ = 4G,

9KG

6 K + 2G

K, =

Y~ 2(1 -- ~)

Compressional modulus

Y~ G, = 2(1 + ~ 4K,G~

K , + G~

Van Wazer found his surface Young's modulus to be lower than the surface shear modulus and ascribed this to the fact that the Young's modulus was measured by a static method whilst a dynamic one (the resonance frequency method) was used to obtain the surface shear modulus. However, another explanation also seems possible. I t can be seen from Eq. [23] that a Y~ smaller than G~ implies a negative value of Poisson's ratio, or, in other words, a lateral contraction instead of expansion on compression of the film. This is likely to have happened under van Wazer's experimental conditions where the film would not seem to have had complete freedom for lateral expansion. The more important relationships for three and for two dimensions are tabulated for comparison (Table I). These formulas were derived assuming an isotropic material. I t has been shown (12), however, that protein films at higher compression m a y be anisotropic. I n this case there would be more than two fundamental elastic constants. The treatment, however, would again be the two-dimensional analog of three-dimensional elastic theory. A mathematically more rigorous treatment of the general n-dimensional case of which the three- and two-dimensional ones are special cases only has been published elsewhere (13). This work was carried out as part of a research project on the surface chemistry of wheat proteins. REFERENCES 1. FOURT, L., AND HARKINS, W. D., J. Phys. Chem. 42, 897 (1938). 2. ALEXANDER, A. E., AND JOHNSON, P., "Colloid Science," p. 496. Clarendon Press, Oxford, 1950.

ELASTIC MODITLI IN MONOLAYERS

507

3. 1VIOUQUIN,I-I., AND RIDEAL, E. K., Proc. Roy. Soc. (London), All4, 690 (1927). 4. CUMPER, C. W. N., AND ALEXANDER, A. ]~., Australian J: Sci. Research, Set. A, 5, 189 (1952). 5. VAN WAZER, J. R., J. Colloid Sci. 2, 223 (1947). 6. LANGMUIR, I., AND SCHAEFER, V. J., J. Am. Chem. Soc. 59, 2400 (1937). 7. T_&CHIBANA,T., AND INOK~CHI, K., J. Colloid Sci. 8, 341 (1953). 8. FOURT, L., J. Phys. Chem. 43, 887 (1939). 9. TI~APEZNIKOV,A. A., Doklady Akad. Nauk S.S.S.R. 63, 57 (1948). 10. CUMPER, C. W. 1'~., AND ALEXANDER, A. E., Trans. Faraday Soc. 46,235 (1950). 11. SEARLE,G. S. C., "Experimental Elasticity," 2nd ed., p. 10. Cambridge University Press, Cambridge, 1933. 12. HtTGHES, A. H., AND I=~ID]~AL,]~. K., Proc. Roy. Soc. (London) A137, 62 (1932). 13. TSCHOEGL,•. W., Australian J. Phys. 11, 154 (1958).