3 Equilibrium Film Pressure on a Flat, Low-Energy Solid R O B E R T J. G O O D Department of Chemical Engineering, State University of New York at Buffalo, Buffalo, Ν. Y. 14214
Introduction The equilibrium f i l m pressure of an adsorbate, πe, i s defined as πe
=
Y
S -
Y
SV
(1)
where ΥS i s the surface free energy of the s o l i d i n vacuum, and YSV i s the surface free energy of the s o l i d i n contact with the saturated vapor of the ad sorbate. The value of this property for an adsorbate which, as a l i q u i d , forms a non-zero contact angle on the s o l i d , has been a matter of uncertainty for some time (1-9). This fact has detracted from the use fulness of measurements of contact angle, θ, for the estimation of s o l i d surface free energies (2,3). See r e f s . (5) and (6) for an important discussion of the problem as it has existed in recent years. The measurement of πe i s not p a r t i c u l a r l y easy; and up to very recently (8) the only determinations that had been reported for low-energy s o l i d s were made on powders (4,5,6), while reported contact angle measurements were made on e s s e n t i a l l y f l a t surfaces. Fox and Zisman (1) found reason to conclude that πe, i s probably n e g l i g i b l e ; and this assumption i s basic to t h e i r "Yc" method of t r e a t i n g contact angle data. Adamson (3) has pointed out that the existence of a t h i c k adsorbed f i l m , and consequent nonn e g l i g i b l e value of πe, are compatible with the e x i s tence of a non-zero contact angle, provided the ma terial in the adsorbed f i l m has a structure that i s s i g n i f i c a n t l y d i f f e r e n t from the bulk l i q u i d . He hypothesizes a degree of "structure" i n a m u l t i l a y e r f i l m , which decays with distance from the surface. If such t h i c k , structured multilayers e x i s t , the low 28
3.
GOOD
Equilibrium
29
Film Pressure
entropy associated with such a structure could account for the existence of a non-zero contact angle. But of course, one cannot argue that the existence of a non zero contact angle proves the presence of a thick, structured f i l m . And indeed, i t i s hard to imagine that a m u l t i l a y e r f i l m of, say, CCl^ on Teflon could have a "structure that would meet the requirements noted i n Ref. 3. Whalen (6) pointed out that the Hill-deBoer equation (11,12,13), with empirical con stants obtained from measurements with hexane and octane on Teflon TFE i n the low-coverage region, pre d i c t s a submonolayer l i m i t i n g adsorption at saturation. The purpose of this paper i s to develop a method of estimating 7r , a p r i o r i , on a molecularly f l a t , homogeneous, low-energy surface, f o r adsorbates of l i q u i d s f o r which θ > 0, without introducing any physical i n t e r a c t i o n s other than those f o r which quantitative expressions are a v a i l a b l e i n well-known physical theory (10). As a f i r s t step i n this a n a l y s i s , an attempt was made to compute, by methods which w i l l be described below, the constants for a BET m u l t i l a y e r adsorption isotherm for CCI/, on a homogeneous, molecularly smooth fluorocaroon surface. I t was found that the BET "c" constant was very small. I f the adsorbent were a powder instead of a f l a t surface, a very low value of c would indicate a BET type I I I isotherm, the f i n a l up-turn of which (near ρ = p ) was due to enhanced adsorbate-solid i n t e r a c t i o n s at points of s o l i d - s o l i d contacts, e.g. as pendular r i n g s , which Wade and Whalen (.5,(5) have discussed. Such s t r u c t u r a l features are, by d e f i n i t i o n , absent from a molecularly smooth surface. These preliminary r e s u l t s led us to turn to a Langmuir model, as being f a r easier to treat than a BET model. It was a n t i c i p a t e d that a s t r i c t l y Langmuirian model might break down when some possible values of molecular parameters were assumed — for example, the model might p r e d i c t high enough coverage that l a t e r a l i n t e r a c t i o n s would be important. For such a regime, a Hill-deBoer isotherm would be more s u i t a b l e . However, i n the present computations, we confined that aspect of the study to the estimation of the conditions under which the Langmuir postulates break down. To a n t i c i p a t e some of our r e s u l t s , a breakdown of the Langmuir assumptions was i n f a c t found, and i n a very i n t e r e s t i n g region of the range of chain-length for n-alkanes, on Teflon. 11
e
Q
ff
11
30
ADSORPTION
AT
INTERFACES
Theory Consider the low-coverage sub-monolayer region of an adsorption isotherm on a uniform s o l i d . We w i l l estimate the energy and entropy of adsorption, with the bulk l i q u i d as the reference state. The p a r t i a l molal entropy of the adsorbed molecules, i n a Langmuir adsorbed f i l m on a s o l i d , i s given by (14). S - - Rln [ x / ( l - x ) l a
(2)
a
where x i s the "θ" of the Langmuir adsorption equa t i o n , i . e . the mole f r a c t i o n of surface s i t e s that are occupied. This entropy term i s c o n f i g u r a t i o n a l ; i t arises from the p o s s i b i l i t y of permuting the molecules among the occupied and vacant s i t e s . The p a r t i a l molal entropy of transfer, from bulk l i q u i d to ad sorbed f i l m , i s given by, X W AS = - Rln τ — + Rln ^r- + AS. . - S (3) l-x W internal ql ' The l a s t two terms i n Equation 3 can probably be neglected. These correspond, r e s p e c t i v e l y , to the changes i n i n t e r n a l , molecular degrees of freedom, and to the configurational entropy of the l i q u i d , considered as a q u a s i - l a t t i c e with occupied and vacant s i t e s . The l a t t e r term i s small, probably less than 0.5 entropy unit (15). The second term on the r i g h t involves the number of angular configurations accessible to a molecule i n the l i q u i d (W^) and i n the adsorbed state (W ). For a quasi-spherical molecule such as C C I 4 , this term i s zero. For an elongated molecule, such as an n-alkane longer than propane or η-butane, we may estimate this term as follows: The molecule i s treated as a r i g i d c y l i n d e r , of length I and diameter d. In the l i q u i d , i t s axis may be at any angle, r e l a t i v e to a fixed coordinate system; i n e f f e c t , i t has a volume 0.75ΤΓ(Ί/2)3 accessible to i t . In the adsorbed state, the energy of a t t r a c t i o n between the s o l i d and the extended molecule renders i t improbable that the molecule w i l l have i t s axis i n any plane that i s at an appreciable angle to the surface. So i t may be r e garded as having, accessible to i t , a volume 2ir(l/2)^d. This entropy term, then, i s approximately Rln(3d/4^). A further refinement on t h i s term can be made by counting the numbers of bent configurations e x p l i c i t l y . These are tedious to enumerate, but t h e i r e f f e c t on the equilibrium w i l l be to predict even lower coverage than that estimated with t h e i r neglect. a
Q
Q
Ί
Ί
v
a
L
a
3.
Equilibrium
GOOD
31
Film Pressure
Thus, we can write as a reasonable approximation, f o r the entropy of transfer per mole between l i q u i d and adsorbed state: N k t l n ^ J - - ln(ff)] (4) a where Ν i s the Avogadro number. For symmetrical ad sorbate molecules, the l a s t term i n the brackets i s omitted. Volume changes are small, i n transfer from l i q u i d to adsorbed state, so we may estimate the enthalpy change as follows: The p a r t i a l molal energy change, i n transfer from the pure l i q u i d to the adsorbed state, i s : AS = -
Δϋ « AU
V
- ÂÏÏ
ads
(5)
V
Here, AU i s the molar energy of vaporization, and AU the p a r t i a l molal energy of adsorption from the vapor. For the purpose of a s t r i c t l y Langmuir-type computation, we assume A U to be constant. Estimates of A U have not been b r i l l i a n t l y successful i n the past; but we are not d i r e c t l y interested i n the heat of adsorption from the gas. Rather, we are interested i n the energy of transfer from the l i q u i d . Thus, i f we use a model that i s moderately reasonable to estimate AU , and the same model to estimate AU * , we w i l l have a good probab i l i t y of making a better estimate of the difference between these two terms. It w i l l develop that we use the macrospic heat of vaporization of the pure l i q u i d as our experimental parameter i n p r e d i c t i n g coverage. Molecular considerations w i l l enter only i n regard to estimating the energy of i n t e r a c t i o n between unlike molecules or segments, and to taking molecular shape into account. Thus, we can write, for substance i , taking the l i q u i d as the reference state, a d s
a d s
acis
V
ac
A
U
N
€
z
/
s
2
( 6 )
i " ii iL Here, ζΐχ i s the coordination number for substance i i n the l i q u i d , and €±± i s the energy at the minimum of the pair p o t e n t i a l function, f o r substance i . Let the molecules of the l i q u i d be designated 1, and those of the s o l i d 2. Then the energy change per mole, on transfer of molecules of type 1 from the l i q u i d to the adsorbed state, i s : Δϋ ^ N < *
1 L
€ /2 u
- «
l a
€
1 2
/2)
(7)
ADSORPTION
32
AT
INTERFACES
Here €^2 i s the energy at the minimum of the p o t e n t i a l function, f o r 1-2 bimolecular i n t e r a c t i o n , and z ^ i s the number of nearest neighbor surface molecules, or groups, that can i n t e r a c t with a molecule of adsorbate, The l a s t term the brackets i n Equation 7 corresponds to the energy of adsorption from the vapor, estimated according to the same model used f o r Equation 6· Equating Δϋ and TAS, we obtain a
-
NkTUn^-H
ln(|f)] = N ( N Z
1L
Z l L
€ n
flafl2
11
ζ
z
l a
€
1 2
)/2
(8)
1ΐΛΐ
I f the dominant types of a t t r a c t i v e forces f o r the two species are the same, i t has been shown that 6^2 *- given, approximately, by ( 1 0 ) : s
(9)
'12
The r i g h t side of Equation 8 then becomes: Ν
ζ
η Λ ι
(10)
τ
I f the coordination number of a molecule (or group) of type 1, i n the l i q u i d , z ^ , i s the same as that of a molecule (or group) of type 2 i n i t s l i q u i d state, ζοτ, and i f there i s the same degree of v a l i d i t y f o r trie assumption of pairwise a d d i t i v i t y of energies f o r substances 1 and 2, and the same f r a c t i o n a l contribution due to neighbors outside the f i r s t coordination s h e l l , then, L
€
ll
/ €
22
Ξ
(ID
AUJf/AUj
Combining Equations 6, 10 and 11 with Equation 8, we obtain: log1-x.
2.3RT
^la
1 Z
(12)
1 L VAUj
A simple case, to which we can apply Equation 12, i s C C I 4 on polytetrafluorethylene (Teflon TFE). We estimate the r a t i o of energies, ^22^11 > a
s
3.
A U
Equilibrium
GOOD
£C1//
a u
$FO
β
Film Pressure
F
o
r
t
h
i
s
P
33
u r
P°se,
A U
ÏFo
2
i
s
estimated
z
from ^heat of vaporization data for CF4 (16). Using the experimental energy of vaporization f o r CCI4, i t i s found that χ = 3 χ 10"^. The f i l m pressure f o r a d i l u t e monolayer can be computed from s,™ x 7Γ (13) e l-x o
σ
a
where a i s the area per molecule i n a close-packed monolayer. For CCIA on Teflon, assuming σ = 30A , the r e s u l t i s ir = 4 χ 10"^ ergs/cm . Thus we obtain a f i r s t , important conclusion: For one l i q u i d that forms a non-zero contact angle on a low-energy s o l i d (!) > ^e should be n e g l i g i b l e , provided the s o l i d i s homogeneous and molecularly smooth. An important system to which t h i s theory must be applied i s the series of homologous n-alkanes on polytetrafluoroethylene. For t h i s purpose, we must modify Equations 8 to 12. We may assume the T e f l o n surface to consist of extended chains. The zigzag structure of the fluorocarbon chain j u s t matches the period of zigzag i n a saturated hydrocarbon chain. We may neglect the h e l i c a l configuration of the f l u o r o carbon, because the p i t c h i s small, about 14 carbons for a turn of the h e l i x . The fact that the l a t e r a l spacing between ( C F 2 ) n chains i s wider than between ( C H o ) chains does not a f f e c t t h i s computation. For an n-alkane, the methyl groups must be treated separately from the methylenes, because the p o l a r i z a b i l i t y of a C H 3 i s considerably greater, and a terminal C H 3 w i l l have more nearest neighbors i n the l i q u i d than w i l l a mid-chain C H 2 . For a long-chain hydrocarbon, most of the neighbors of a C H 3 i n the l i q u i d state are C H 2 groups, so the energy of i n t e r a c t i o n between C H 3 groups i n the l i q u i d may be neglected. Then Z - Q ^ H , Equation 7, may be replaced with the expression Z
n
^ LCH2
and z
l a
( n
€
"
1 2
2 ) z
CH2 9
L
C
H
^
L
C
H
^
,
^
^
, by €
aCH CH ,CF 2
2
+ 2
2z
G
aCH CH ,CF 3
3
With t h i s model, the expression i n Equation 10, becomes
2
for AU,
(15) employed
34
ADSORPTION
Δϋ - f
+ 2z
^
^ LCH^ CH^CH^
LCH,
CH2CH2
CH^CH^
1 -
'aClk
"CF CF
'LCH.
•CH C
2
0
2
C
H
'aCH.-
-CF CF
'LCH,
:
2
A T INTERFACES
2
2
2
(16)
CH2 CH„
We can estimate the 6's for methylene and methyl groups i n terms o f energies of vaporization, as was done f o r use i n Equation 1 2 . As before, we approximate the r a t i o s under the square roots, by the r a t i o o f energies of vaporization of CF4 and CH^. Combining the r e s u l t with the entropy terms, we obtain: /Δυ
'aCR-, log-
1-x
1
(η-2)Δυ;
2.3RT
'LCH,,
«2
2 y
A
U J
,
AU,
CH,
1
CH„
A U
CH
4
i
/Δυ,
'aCH, +
CF.
-
CF.
- log(|f) ( 1 7 )
-
Δϋ,CH.
'LCH-
We have c a r r i e d out computations f o r n-alkanes on Teflon, using A U C H 3 2 5 0 joules/mole and ΔΗ^Ηβ = 8 2 0 0 joules/mole, from vapor pressure data ( 1 6 ) , and 2
Z
/ z
2
/
4
=
z
/ z
1
,
1
T
h
e
r
a
t
i
o
d
/
l w
a
s
aCH LCH " ' aCH LCH = ' estimated assuming the van der Waals diameter of a n-alkane to be 4.8A, and the projection of the C-C distance on the chain axis to be 1.252A. Table I shows the r e s u l t s of this c a l c u l a t i o n . Values f o r pentane and butane are i n parentheses because the assumptions that most of the neighbors of a C H 3 group are C H 2 groups, and that the molecule l i e s f l a t on the surface, break down f o r short-chain alkanes. We note at once, from Table I, that x and are e s s e n t i a l l y zero f o r the higher hydrocarbons. Below hexane, χ increases r a p i d l y ; and the assumption of no l a t e r a l interactions quickly becomes inapplicable. So this computation leads d i r e c t l y to a p r e d i c t i o n of the region where the method of computation should break down. Thus, i t 2
2
3
3
a
3.
Equilibrium
GOOD
Film Pressure
35
Table I. n-alkane
2
X
octadecane
1.4 χ
a 10~
hexadecane
4.2 χ
decane
1.1 χ
TT ,ergs/cm e
6
4.4 χ
10~
5
10"
6
1.5 χ
10"
4
10~
4
5.6 χ
10"
3
10"
2
octane
4.0 χ
10"
4
hexane
1.7 χ
10"
3
pentane
(2.9 χ
butane
(5.4 χ
2.3 χ 0.12 3
(0.23)
3
(0.48)
10" ) 10" )
i s as expected, that the computed values f o r butane and pentane are much below those observed (4). It i s possible that the estimates of z çHo d aCH3 used above may, f o r c e r t a i n kinds of s i t e s , be too Small. For a r e a l surface, i t i s highly probable that step or ledge s i t e s e x i s t , for which z cHo y ke 3 and z Q^ may be as 5. Such s i t e s could w e l l constitute 5 or 10% of an experimental surface. Examination of models shows that, for elongated molecules, i t i s u n l i k e l y that z cH2 l d be as large as 3 f o r the e n t i r e chain, together with z çH2 s large as 5 for every terminal C H 3 . Indeed the values of these two z's w i l l depend on chain length, and on the l a t e r a l spacing of fluorocarbon chains. I t i s only for convenience that we approximate them by constants, and we w i l l assume average values of 2.5 f o r z çjj2 * 4 for z ç7io. An aiicane molecule i n a s i t e such as j u s t described w i l l not have freedom to take up any angular o r i e n t a t i o n p a r a l l e l to the plane of the surface; indeed there w i l l be just two orientations which have the same energy. So the term, log (3d/4t), i n Equation 17 must be replaced by log (3d2/£ ) Table II shows the r e s u l t computations of this type: The f r a c t i o n x ( tep}°f step s i t e s on Teflon TFE covered by n-alkane molecules, and the contribut i o n to the f i l m pressure due to t h i s coverage, assuming 5% of the surface i s accounted for as step sites. a n
a
z
m a
a
a
c o u
a
a
a
a n c
a
a
2
e
a
s
36
ADSORPTION
AT
INTERFACES
Table I I : Estimate of f r a c t i o n a l coverage of step s i t e s on t e f l o n TFE by n-alkane molecules, and c o n t r i b u t i o n of t h i s coverage to equilibrium f i l m pressure, assuming 5% of surface i s accounted f o r as step s i t e s . Table II n-alkane octadecane
χ , a(step) 8.5 χ 10
hexadecane
1.8 χ 10
fc
0.05ττ , ergs/cm e' 1*3 χ ΙΟ"
N
4
3
3.2 χ 10
decane
2.2 χ 10~
2
octane
5.2 χ 10"
2
hexane
1.5 χ 10"
1
(pentane)
2
5.8 χ 10"
1.5 χ 10 "
1
0.50 1
(0.8)
1
(1.3)
(2.0 χ 10" )
(butane)
2
&
(3.0 χ 10" )
The values f o r pentane and butane are given i n paren theses because of the breakdown of an assumption ( i n addition to those noted regarding Table I) that was made i n the s t e p - s i t e model. This i s , that the aver age values of ζ , τ τ and z ~ are less than 3 and 5, respectively. 2 3 We note at once, from Table I I , that the comput ed contributions to ir f o r the higher n-alkanes i s not a p p r e c i a b l e — j u s t as was seen i n Table I. And for the lower alkanes, f o r which the model breaks down, the computed contribution to ir i s s t i l l r e l a t i v e l y small — only 1.9 erg/cm2 for butane, for which i t i s computed that about 30% of the step-sites are occupied. F i n a l l y , we w i l l treat water on Teflon and on polyethylene, using Equation 12 with omission of the term, log(3d/4£). To evaluate ^^_2 * relations : ο Γ
u
a C H
a C H
e
y w
€
d
+
"
+
6
μ
e
u
s
=
e
t
le
" C " )
Κ
1
+
B
«A
+
1
~TET
3.
GOOD
Equilibrium
e
2 2
Film Pressure
= ^22
fc
12
Here
37
*12
(
J4l
€
22
2
0
)
21
< >
i s computed by the equation
*
d
22
- -r+M-
< >
where I i s i o n i z a t i o n energy. These r e l a t i o n s are discussed i n d e t a i l i n Ref. (10). The superscripts d, i and μ,, i n Equations 18 - T 2 r e f e r r e s p e c t i v e l y to dispersion, induction and permanent dipole components of intermolecular energy. α i s p o l a r i z a b i l i t y , I i s i o n i z a t i o n energy, μ i s dipole moment, and Β i s a constant which, f o r molecules c o n s i s t i n g of atoms i n the second and t h i r d rows of the periodic table, i s close to 0.66 (10). For water, the r a t i o computed by Equation 19 i s close to 0.20 (10). Equations 18 and 19, are i m p l i c i t i n the discussion of Fowler and Guggenheim (17,18). Equation 21 can be put i n the form, e
"- "fe-% €
€
- 11
° '
2
v
(23>
< > 24
The reduction i n average coordination number, ùz , f o r water, on going from bulk l i q u i d to adsorbed state, i s probably about 1, i . e . from something near 4, to about 3. Polyethylene i s treated as extended (CH2) chains. The r e s u l t s are given i n Table I I I . It should be emphasized that these computations are f o r water on i d e a l Teflon TFE and polymethylene surfaces, i . e . assuming the s o l i d s to be smooth and homogeneous* n
38
ADSORPTION
AT
INTERFACES
Table I I I . Estimates of f r a c t i o n a l coverage and of equilibrium f i l m pressure f o r water on a fluorocarbon s o l i d and on a polymetheylene surface, both assumed to be smooth and homogeneous. Solid
χ a
7Γ , ergs/cm e 7-
Teflon TFE
6 χ 10
2.5 χ 10
Polyethylene Discussion
7 χ 1θ"
6
3 χ 1θ"
5
(a) The Approximations. We must now make a c a r e f u l examination of the important approximations that we have introduced. The f i r s t , and possibly the most important, approximation i s the use of the Langmuir equation i t s e l f . The very low values of x computed show that these systems (except for the lower hydrocarbons) should be within the coverage region where the Langmuir equation can s a f e l y be used. We have, of course, neglected l a t e r a l interactions so far; and we now examine the v a l i d i t y of that neglect. For a quasi-spherical molecule such as CCl^, the energy term — i . e . the r i g h t side of Equation 12 — becomes a
ΔΙΙν
^la
lia
273RT
5
IL
AU;
5
(25)
1L
where z"£ i s the average number of " l a t e r a l " nearest neighbors of an adsorbed molecule. ^ In a perfect 2-dimensional, hexagonal l a t t i c e , z'i would be 6. For CH2 groups i n an n-alkane, z £ i s at most 2 . The entropy of adsorption w i l l be less than the Langmuir entropy (14), so to use the expression ,(25) instead of the corresponding term i n Equation 1 2 w i l l y i e l d the maximum value of x : a
v
a
a
a
l-x-
exp
ΔΙΙ JL RT
'ΔΙΚ
1 - 'la 1L
'la S
5
(26)
1L
For C C I 4 on Teflon TFE, with zî /ziL =0.5, this treatment y i e l d s x < 10" . Thus, the refinement of computing x^ with allowance f o r l a t e r a l i n t e r a c t i o n s does not bring the estimated coverage s e r i o u s l y outside the region of the isotherm where l i n e a r i t y i s a
2
a
3.
GOOD
Equilibrium
Film Pressure
39
expected. Hence, the use of the Langmuir isotherm as an acceptable approximation, f o r molecules such as C C I 4 , appears to be j u s t i f i e d . (For step s i t e s , the s i t e s themselves w i l l be i s o l a t e d from each other, and so there i s no l a t e r a l i n t e r a c t i o n of adsorbate molecules even when most of the step s i t e s are covered). For l i q u i d s having much lower b o i l i n g points than CCl^, however, i t i s to be expected that l a t e r a l interactions and two-dimensional condensation w i l l be important, and hence that larger values of ττ w i l l be found. A second approximation i s the assumption of chemical uniformity of the s o l i d surface. It has been established (19) that hydrophilic s i t e s e x i s t on at l e a s t some, ancT possibly a l l , samples of Teflon TFE. Such s i t e s are, no doubt, chemically d i f f e r e n t from the majority of s i t e s . On polyethylene, oxygencontaining groups are l i k e l y to be present at the surface; and on other low-energy surfaces, i t i s to be expected that high-energy heterogeneities should commonly be present i n a f i n i t e concentration. Such s i t e s would i n t e r a c t with water molecules, and could contribute to a large value of ττ as computed from water adsorption, yet not make an excess c o n t r i b u t i o n to hydrocarbon adsorption. Heterogeneity w i l l also be present for geometric reasons, such as microscopic roughness or microporosity beyond the l e v e l of the step-sites considered above. So some adsorbed molecules w i l l have more nearest neighbors, i n terms of groups such as C F 2 , C F 3 etc. on Teflon, than w i l l others. But i n general, i t i s very u n l i k e l y that z-^ w i l l be larger than 5 or 6. So the r a t i o z^ /z-n w i l l be, at most, about 0.5; and the large values or this r a t i o w i l l pertain to only a very small f r a c t i o n of surface s i t e s , on a surface which i s smooth enough for contact angle measurements to be made. The r a t i o , ^ 2 2 ^ 1 1 ' ^ l° " §y s o l i d , 2, i n contact with any l i q u i d , 1, w i l l seldom be very large; for a hydrocarbon on a fluorocarbon, i n the computations given above, i t turned out to be at most 1.25. There w i l l be no more than a small f r a c t i o n of the s i t e s for which the energy term i n Equation 12 or 17 i s , for geometric reasons, very much larger than that computed above. And f o r elongated molecules on such s i t e s , i t has already been noted that the angular entropie term (the l a s t term i n Equations 12 and 17) w i l l be much more negative, because to occupy such a high-energy s i t e , an extended molecule w i l l have at most two configurations access i b l e to i t . Hence, the coverage of an e s s e n t i a l l y Β
Θ
a
a
o r
a
w
e n e r
40
ADSORPTION
AT
INTERFACES
f l a t surface w i l l not be s e r i o u s l y l a r g e r than that c a l c u l a t e d here. Of course, surfaces can be prepared which are very much more heterogeneous, f o r geometric reasons, than j u s t indicated, e.g. by abrasion. But as pointed out by Neumann and Good (20), such surfaces are not suitable for contact angle measurements ; and the r e p r o d u c i b i l i t y of data obtained would be very poor, and the hysteresis extremely large. Also, the heterogeneity of a surface that has been modified, by p a r t i a l oxidation or by g r a f t i n g short hydrophilic chains onto i t , w i l l be much more serious than that of an unmodified surface. The f i l m pressure on such surfaces i s a very d i f f e r e n t problem from that on an e s s e n t i a l l y homogeneous, low-energy surface, and i t w i l l not be discussed here. The term Rln(W /WL) *rigorous expression f o r the change i n entropy associated with angular conf i g u r a t i o n s . The approximation of evaluating i t as Rln(3d/4£) has already been discussed. The error thus introduced r e s u l t s i n p r e d i c t i o n of too high an adsorption; so the conclusion as to the n e g l i g i b l e value of ir f o r higher hydrocarbons i s not weakened by i t . Regarding the expressions containing the €'s, e.g. Equation 10, i t may be seen at once that € ^ i s by f a r the most important parameter, because the r a t i o , la/ 1L generally small, e.g. 3/12, or r a r e l y larger than perhaps, 5/11. Since i s evaluated from AUÏ, i t i s c l e a r that the dominant energetic component of the computation i s the heat of vapori z a t i o n of the l i q u i d . There i s considerably less energy than A U "recovered , on adsorption, because the coordination number i s so much lower, f o r the adsorbed molecule with the groups i n the adsorbent surface. The estimation of €^2 from >/€^^Z2 3 i ° 9, i s i f anything, on the high side, even f o r systems where the cohesion of the bulk l i q u i d and of the s o l i d are both dominated by the London force. A more exact expression, replacing Equation 9, i s s a
a
e
Z
Z
i s
V
11
9
€
1 2 = * - ^ Λ 2 "
E
where Φ can be computed a p r i o r i , from molecular properties (10), e.g., by Equation 22 f o r nonpolar molecules. The fact that we express the €'s i n terms of A U s (e.g. Equations 11 to 17) i s , i n a p r a c t i c a l V f
3.
GOOD
Equilibrium
Film Pressure
41
sense, a strong point of this theory. We have already noted the assumptions m a d e — s e e the paragraph preceding Equation 11. The main factor leading to inequality of z^j and (see above) i s , differences i n decree of expansion or the l i q u i d s , considered as f r a c t i o n of q u a s i - l a t t i c e s i t e s that are vacant, at the temperatures where the heats of vaporization are measured. Since the b o i l i n g point i s , to a f a i r approximation, a "corresponding temperature within the meaning of the theory of corresponding states, we can conclude the equality of coordination numbers i s a good approximation. Even i f these errors do not cancel t o t a l l y , they should do so p a r t i a l l y . And since this r a t i o appears under a square root sign, Equation 12, and then i s m u l t i p l i e d by a factor that i s of the order of 0.25, the s e n s i t i v i t y of the theory to errors of this kind i s small. Equations 16 and 17 introduce approximations needed to treat highly asymmetric molecules. The asymmetry cannot be handled i n terms of any e x i s t i n g , macroscopic treatments which use heats of vaporization d i r e c t l y . These assumptions are, however, a p r i o r i , reasonable, and so can be counted on as leading to a v a l i d p r e d i c t i o n of the trends. Equations 18 to 22 are based on Good and Elbing (10) and (as already noted) on the e a r l i e r treatments of Good and G i r i f a l c o (2.,Z>18) and Fowler and Guggen heim (17). A notation basecPon that of Fowkes (21) has been employed; but the quantities are not obtain able from empirical treatment of surface tension data, as i s the case with Fowkes Y '. Indeed, the dispersion component of the t o t a l surface energy can not i n general be evaluated from contact angle data, because of the lack of thermodynamic uniqueness of the U function, i n a binary system. (It i s only when there i s no i n t e r f a c i a l excess mass of e i t h e r . component that U i s unique. In general, U ( '£ U ( ), where the superscript designates the Gibbs d i v i d i n g surface: (1) r e f e r s to the surface located such that Γ\ = 0, and (2), to the Γο = 0 surface.) Thus, i s a function which has physical meaning only i n terms of the components of intermolecular a t t r a c t i o n constants as computed from molecular properties, e.g. by Equation 18 and 19. The c o e f f i c i e n t , 0.2, i n Equation 24, arises from t h i s treatment of the energy components f o r i n t e r a c t i o n of water and a nonpolar s o l i d . I t s use i n the theory for water depends f o r accuracy, not so much on the v a l i d i t y of Equation 18 or 19, as on the assumptions 11
1
lf
df
s
s
S
L
s
2
42
ADSORPTION
AT
INTERFACES
about coordination number and second-nearestneighbors, which j u s t i f y the use of the r a t i o of^ energies of vaporization i n Equation 24. We estimate the uncertainty here, as no worse than 50% i n ^i\l^22 which means an uncertainty that i s less than 25% a f t e r the square root i s extracted. The q u a l i t a t i v e conclusion that x and rr are n e g l i g i b l e , f o r H2O on Teflon and polyethylene, are not s e r i o u s l y changed by allowance for such uncertainty, because A U i s so large, f o r water, and as a consequence, the computed values of x and ΊΤ are so small. In summary, the approximations of t h i s theory are probably not serious; and indeed, the majority of the approximations contribute errors i n the d i r e c t i o n of too high an estimate of x and ττ · It would take very much greater c o r r e c t i o n factors than now seem l i k e l y to appear, i n any refined theory, to change the q u a l i t a t i v e conclusion that x and ττ are generally small for l i q u i d s that b o i l above room temperature. It must be emphasized, however, that these compu tations are not intended as quantitative predictions of x arri 7r for these systems. The approximations made above, including those about structure of the s o l i d surface, are s u f f i c i e n t l y serious that quanti tative agreement cannot be expected. But for the p r e d i c t i o n of trends, as with the n-alkanes, and for the conclusion that ττ i s small for high b o i l i n g l i q u i d s , the approximations should not detract from the v a l i d i t y of this theory. (b) Comparison with Experiments: i r for Organic Compounds on Teflon. Graham (4) has found ir to be much smaller, f o r η-octane on powdered Teflon TFE, than for alkanes having surface tension below Y , e.g. butane. ir has been reported to be large f o r low-boiling gases such as N on Teflon. Graham (4) noted a d i f f i c u l t y a r i s i n g from bulk s o l u b i l i t y of an alkane i n the substrate, such that the quantity sorbed could not be uniquely assigned as between adsorption and absorption. (This trouble was not encountered by Whalen and Wade (5,6.)). Graham concluded that his value of ir for octane on Teflon, about 1.7 erg/cm^, was an upper l i m i t ; he could not estimate the r e a l value. Wade and Whalen (5,j5) also used a powder form of Teflon, and also oBtained a small value for i r : 3.3 f o r hexane and 2,9 for octane. They e x p l i c i t l y corrected for pendular r i n g condensation, and i n so doing, they obtained estimates of ir i n t h e i r systems that were very probably more v a l i d than Graham's were f o r h i s . Our computations are lower than these experimental r e s u l t s y
a
e
V
a
a
θ
a
a
e
e
e
c
e
2
e
e
e
3.
Equilibrium
GOOD
Film Pressure
43
e.g. ττ =0.1 f o r hexane i n the absence of step s i t e s . The r e s u l t , 7 r =0.5 for Teflon with step s i t e s accounting for 5% of the area, shows the d i r e c t i o n of change i n the computed r e s u l t s when a more complex surface i s postulated. The preliminary computations made f o r CCl^ indicate that the e x p l i c i t employment of a two-dimensional van der Waals equation (11,12) would be worthwhile with the lower hydrocarbons. We w i l l investigate such two-dimensional condensation i n another communication. We note, however, that our equations have successfully predicted the observed trend of ir with chain length. As has already been noted, water adsorption measurements at Lehigh (19) have shown that the surface of Teflon powder i s heterogeneous, and about 0.757 of the surface of the sample studied consisted of s i t e s which adsorbed water strongly. The computed adsorp t i o n of water, reported i n Table I I I , i s f o r the 99% of the surface exclusive of the hydrophilic s i t e s . The Lehigh measurements (19) could not q u a n t i t a t i v e l y d i s t i n g u i s h the increment of water adsorption due to t h i s f r a c t i o n of the surface. The water isotherm was described as resembling a type II isotherm, with surface area less than 1% of the nitrogen area, superimposed on a type I I I isotherm. We have not attempted to analyse the contribution to the adsorption of hydrocarbons or other nonpolar molecules, due to the hydrophilic s i t e s on Teflon. Wade (6) has discussed the evidence f o r interactions of high-energy s i t e s on Teflon with hydrocarbon molecules. Whalen (22) has recently measured the adsorption of water on Teflon powder, and estimated a value of 1.9 ergs/cm for ?r . He estimates the hydrophilic s i t e s on this s o l i d to account f o r 3%> of the surface area (6,23). Adsorption measurements on Teflon powders are of l i m i t e d d i r e c t pertinence to the contact angle question because there i s every reason to suspect that the surface of a powder i s quite s i g n i f i c a n t l y d i f f e r e n t from that of a smooth slab that i s suitable for contact angle measurement. This point i s hard to v e r i f y d i r e c t l y , however, because while electron microscopic examination of a grossly smooth s o l i d could reveal pores, i t would not reveal chemical heterogeneity, p a r t i c u l a r l y that i n the form of i s o l a t e d , high-energy s i t e s ; and the p r e c i s i o n measure ment of contact angle on a powder i s not easy. e
e
o
2
e
(c) Comparison With Adamson's Model, and Results For Water On Polyethylene.
His
44
ADSORPTION
AT
INTERFACES
(1) Theory. Our model i s , i n p r i n c i p l e , not incompatible with that of Adamson and Ling (3), i n that t h e i r isotherm graph, i n i t s lower pressure region, resembles a BET type I I isotherm. I t i s w e l l known that, by lowering the value of the " c " constant, a type I I I isotherm i s obtained which, i n i t s lowcoverage region, i s not experimentally d i s t i n g u i s h a b l e from the Henry's law region of a Langmuir isotherm. Figure 1 shows a schematic graph of the isotherm we propose. Since the coverage at saturation i s f a r below a monolayer, there i s no need to make any hypotheses about the structure of the absorbed mater i a l i n the region above monolayer coverage. (2) Experiment. Adamson, i n 1973, reported (8,9.) a value of 42 ergs/cm f o r ττ of water on polyethylene. These r e s u l t s appear to be i n disagreement with Table I I I . They are also i n apparent c o n f l i c t with the w e l l known conclusion (24) that the Y polyethylene i s i n the neighborhood dF"31 ergs/cm . The use of Equation 28, [Y (cos0+l)- 7Γ 2 ] •Li e Y θ
c
2
T
S
4Λ
\
(28)
Y (cose+i)
2
L
4Φ
2
with Φ about 0.9 to 0.95 (1,10) leads to an estimate of about 36 ergs/cm f o r Yg. I t would be rather anomalous f o r polyethylene i n water vapor to have a negative surface free energy Ygy, which would be obtained by subtracting 42 from"36 as Equation (1) would seem to prescribe. A r e c o n c i l i a t i o n can be achieved by hypothesizing that water on the polyethylene sample that was used (8.,9.) behaved i n a s i m i l a r fashion to water on Teflon TFE, (19) noted above. The water molecules might adsorb i n clusters on i s o l a t e d hydrophilic s i t e s , y i e l d i n g a f i l m with the observed average thickness and the reported ττ^. With respect to water adsorption, the e f f e c t i v e Y g of this s o l i d would then be, not the value derived from contact angles of nonpolar l i q u i d s , but a larger value: 2
Y
S
=
Y
+
S ^e(water) -36+42 , = 78 ergs/cm 2
(29)
3.
GOOD
Equilibrium
Film Pressure
45
Figure 1. Schematic: A possible isotherm for adsorption of a liquid such as carbon tetrachloride or hexadecane on a molecularly smooth, homogeneous solid such as Teflon TFE. Extrapohtions shown on log-log plot are speculation, and no physical reality is intended, particularly beyond the region where the curve starts to swing to the left.
46
ADSORPTION
AT
INTERFACES
1
I f the water c l u s t e r s on Adamson s p o l y e t h y l e n e were i s o l a t e d from each o t h e r , t h e i r p r e s e n c e would not reduce t h e water c o n t a c t a n g l e t o z e r o . And i n the h y d r o p h o b i c r e g i o n s between c l u s t e r s , t h e amount o f water adsorbed p e r u n i t a r e a , and ττ , would be n e g l i g i b l e — a s computed above, f o r T a b l e I I I . β
Conclusions The e q u i l i b r i u m f i l m p r e s s u r e , 7 r , s h o u l d be n e g l i g i b l e on a smooth, homogeneous s u r f a c e o f a lowenergy s o l i d such as T e f l o n , f o r most l i q u i d s t h a t form non-zero c o n t a c t a n g l e s , and p a r t i c u l a r l y f o r those w i t h h i g h h e a t s o f v a p o r i z a t i o n . ir s h o u l d be l a r g e f o r l o w - b o i l i n g s u b s t a n c e s , such as N 2 , ethane, e t c . , on t h e s e s o l i d s . I t i s shown t h a t these p r e d i c t i o n s are i n accord with experimental r e s u l t s . e
e
Acknowledgement T h i s work was s u p p o r t e d by t h e N a t i o n a l S c i e n c e F o u n d a t i o n under Grant GK10602. Literature
Cited
1.
Fox, H.W. and Zisman, W.A., (1950) 5, 514.
2.
Good, R.J. and (1960) 64, 561.
3.
Adamson, A.W., and L i n g , I . , in " C o n t a c t A n g l e , Wettability and A d h e s i o n " , p. 57, Advan. Chem. S e r No. 43, American C h e m i c a l S o c i e t y , Washington, D.C., 1 9 6 4 .
4.
Graham, D.P., J. Phys. Chem., (1965), 69, 4387.
5.
Wade, W.H., and Whalen, J.W., J . Phys. Chem. (1968), 72, 2898.
6.
Whalen, J.W., J . Colloid (1968), 28, 443.
7.
Good, R.J., in " C o n t a c t A n g l e , W e t t a b i l i t y and Adhesion", Advan. Chem. S e r . No. 43, American C h e m i c a l S o c i e t y , p. 74, Washington, D.C., 196 4 .
8.
Adamson, A.W., 4 7 t h N a t i o n a l Ottawa, Canada, June, 1973.
Girifalco,
J.
Colloid
Sci.,
L.A., J. Phys. Chem.,
and
Interface
Colloid
Sci.,
Symposium,
3.
9. 10.
GOOD
EquilibriumFilmPressure
47
Adamson, A.W., 167th National Meeting of the American Chemical Society, Los Angeles, April, 1974. Good, R . J . and Elbing, Ε . , Ind. Eng. Chem. (1970), 62, (3), 54.
11.
de Boer, J . H ., "The Dynamical Character of Adsorp tion," Oxford University Press, London, 1953.
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Ross, S., and O l i v i e r , J . P . , "0n Physical Adsorp tion," Interscience Publishers, New York, 1964.
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Hill, T . L . , "Introduction to S t a t i s t i c a l Thermo dynamics," Addison-Wesley Publishing Co., Reading, Mass., 1960.
14.
Everett, D.H., Proc. Chem. Soc., Feb., 1958, p. 57.
15.
Glasstone, S., L a i d l e r , K.J., and Eyring, Η., "Theory of Rate Processes," McGraw-Hill, N . Y . , 1941.
16.
C.R.C. Handbook, 51st ed., Chemical Rubber Publishing Co., Cleveland, Ohio, 1970.
17.
Fowler, R . H . , and Guggenheim, E . A . , " S t a t i s t i c a l Thermodynamics," Cambridge University Press, London, 1952.
18.
Berghausen, P . E . , Good, R.J., Kraus, G . , Podolsky, Β., and S o l l e r , W., "Fundamental Studies of the Adhesion of Ice to S o l i d s , " WADC Technical Report 55-44, 1955.
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Chessick, J.J., Healey, F . H . , and Zettlemoyer, A . C . , J . Phys. Chem. (1956), 60, 1345.
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Neumann, A.W. and Good, R.J., J . Colloid and Inter face Sci., (1972), 38, 341.
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Fowkes, F . M . , J. Phys. Chem. (1957), 61, 904.
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Whalen, J.W., Vacuum Microbalance Techniques, A.W. Czanderna, E d . , v. 8, p. 121, Plenum Press, 1971.
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Whalen, J.W., (personal communication,
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Zisman, W.A., i n "Contact Angle, Wettability and Adhesion," Advan. Chem. Ser., No. 43, pp. 1-51, American Chemical Society, Washington, D.C., 1964.
1974).
