The States of Monolayers. - The Journal of Physical Chemistry (ACS

20

WILLIAM D. HARKINS AND EDWARD BOYD

THE STATES OF MONOLAYERS1 WILLIAM D . HARKINS

AND

EDWARD BOYD

George Herbert Jones Laboratory, University of Chicago, Chicago, Illinois Received J u l y 3, 1940 I. GENERAL PHASE RELATIONS

Monolayers on a liquid subphase exhibit, in two dimensions in the plane of the surface, properties which justify the classification of their different modifications as solid, liquid, and gaseous phases. At extremely large molecular areas the monolayer acts as a perfect two-dimensional gas, at lower areas as an imperfect gas. The most striking feature which emerges when a general phase diagram is constructed (figure 1) is that there is definite evidence for the existence of a t least two, and possibly three, phases which exhibit the viscosity relations attributed in three dimensions to the liquid state. In table 1 two methods of classifying the two-dimensional phases are presented, together with the probable order of the phase changes. The order assumed for each transition is the subject of a discussion given later. Figure 1 presents the general relations of films whose critical temperaphase is above the lowest temperature for the formation of the liquid (L1) ture of collapse of this phase (at the point P ) , that is, above the temperature at which the two-dimensional system may change to three dimensions. The lowest curve represents the direct condensation of vapor to solid a t So. This may occur, for example, if a film of stearic acid is compressed on an alkaline subphase which contains even a minute concentration of calcium ion. Figure 2 is a general diagram, which supplements that of figure 1 but which may be considered to represent a film, such as that of ethyl pentadecylate, a t temperatures (go, 17", 25", and 32OC.) which lie above the critical temperature for the formation of the liquid L1 film. The exact critical temperature of this film has not been determined but is not far from 9°C. The actual data have been obtained by the use of the four esters ethyl margarate (C1,Et), ethyl palmitate (CleEt), ethyl pentadecylate (ClsEt), and ethyl myristate (C14Et),but an increase of one carbon atom has an effect equal to that produced by a decrease of temperature of 8°C. The detailed relations for these esters are considered in section XIII, but what is important here is that the relations for the transformation gas -+ intermediate, and liquid (L1) -+ intermediate are so closely similar that they cannot be distinguished from each other by the form of the curves. The 1 Presented at the Seventeenth Colloid Symposium, held at Ann Arbor, Michigan, June 6-8, 1940.

21

STATES OF MONOLAYER8

two-phase region (LIG)of figure 1 exists in figure 2 only at higher areas and lower temperatures than those given. Under the proper conditions any one of the five phases may change directly into any other one of the five, except possibly the changes between the liquid expanded and the liquid condensed or solid states, which seem

FIG. 1. General phase diagram for monolayers on a liquid subphase (I). The lowest temperature of collapse a t P is below the critical temperature of the L I phase. G, gas; A , transformation to liquid expanded ( L I )phase begins; LlG, two-phsse region; B, transformation t o L1 ends; BC or Lj, liquid expanded; CD or I , intermediate phase; DE or L2,liquid condensed phase; EF or solid phase. The presence of calcium ions, when the film consists of a long-chain paraffin acid, such as stearic acid, extends FE to a very low pressure, where the transition is between low-pressure vapor and solid film. P represents the collapse of the monolayer t o give liquid lenses: the lower of the points P represents the collapse of a liquid expanded film and the upper that of a vapor film. The temperatures listed on the curves are valid for pentadecylic acid, and should be increased from 8 to 10" for each added carbon atom. With alcohols the temperatures are much higher.

s,

II. FIRST-ORDER TRANSITION FROM A LOW-PRESSURE VAPOR FILM TO EITHER (1) A SOLID, (2) A CONDENSED LIQUID (Lz), OR AN INTERMEDIATE LIQUID FILM (FIGURE 1)

The discussion given above indicates that a low-pressure vapor film of a normal long-chain paraffin acid may, on a basic solution in the presence of calcium ion, condense directly to a solid film at So. Another example of the same type of condensation, according to the literature, is that of the triglycerides.

22

*

WILLIAM D. HARHINS AND EDWARD BOYD

At molecular areas below 44 A.2, figure 1 represents quantitatively the behavior of a monolayer of pentadecylic acid on an aqueous subphase of pH 2; above 44 1.2the area is very much condensed on the z-axis, and the pressure expanded on the y-axis. The purpose for which the strong acid is added is to prevent the formation of calcium and other salts of the longchain acid from the bivalent or trivalent ions almost inevitably present in the water. In a neutral or basic solution a very minute concentration of calcium ion may transform the long-chain acid completely into a salt, TABLE 1 Two-dimensional phases and probable order of the changes between them

1

PEABE

OBDEB OF CEANQQ BQTWQEU PEMFd

Classification A

I. Gas, G. Includes compressed vapors, such as the ‘(vapor expanded” film of Adam

I and 11: first 11. Liquid, LI (BC). “Liquid expanded” Adam. Liquid of high compressibility

of

I1 and 111: resembles diffuse first-order change of phase* 111. Intermediate (I) or transition (CE). Liquid of extremely high compressibility

111 and IV: third

IV. Liquid, L, (DE). “Liquid condensed” of Adam. Liquid of low compressibility

IV and V: second

V. Solid, S (EF) Classification B: same an A, except as indicated below 111. Intermediate: (a) Transition, (b) Liquid, LC

T

IV. Solid

* Possibly

second order; see section XV.

which forms a solid film. This eliminates completely the La film (13). A smaller amount of the bivalent ion may limit the range of existence of the Lz film and decrease its molecular area. At 25°C. the normal long-chain paraffin acids from margaric to arachidic acid (seventeen to twenty carbon atoms), on an acidic subphase, are known (15) to condense directly from a vapor a t a pressure of a few tenths of a dyne to condensed liquid (LI), while palmitic acid condenses from a low-pressure vapor to the intermediate phase, which changes to LZa t a

STATES O F MONOLAYERS

23

pressure.of about 6 dynes em-'. The low-pressure vapor films of the corresponding acids of from fifteen down to thirteen or less carbon atoms condense at this temperature into the L1 or liquid expanded state The quantitative relations for pentadecylic acid (CX) are exhibited in figure 3. This may be considered to be a general diagram for any of the corresponding acids, provided the teniperatures on the curves are given on a Corresponding, instead of an absolute, scale. Thus an increase of

forrt-Arra Curves of lk Ethyl frkrs of the Straight Chain Fatly Acids CliClrmOOIN

hCIat25O.C

FIG.2. General phase diagram (11). The lowest temperature of collapse to give liquid lenses is above the critical temperature of the liquid expanded (Ll) p h y e . This diagram is valid for a single substance a t different temperatures, but the specific curves are those obtained for ethyl esters of margaric, palmitic, pentadecylic, and myristic acids a t 25°C. Figure 6 gives the pressure-area relations for the same esters a t 15°C. The curves for the phase which exhibits the highest areas resemble those for the liquid, L1, state, but the films are gaseous (compressed vapor), with the possible exception of that of the margarate, which is not far from its critical temperature and may be liquid ( L 1 ) . Experiments with the esters should be carried out with a neutral aqueous subphase, preferably carbon dioxide-free water, t o avoid hydrolysis. To avoid the effect of hydrolysis our experiments with the shorter chain esters were completed in an extremely short time.

one carbon atom raises the temperature for any single pressure(a)-area (u) curve by approximately 10°C., at least in the L1 and transition regions. 111. FIRST-ORDER TRANSITION FROM LOW-PRESSURE VAPOR TO LIQUID, Ll. (PHASE I +PHASE 11)

The transition from a low-pressure vapor film to the liquid L1 begins at a high molecular area with the formation of micelles (A, figure 1) of

24


liquid which enlarge to islands as the molecular area is decreased. I n the case of pentadecylic acid this condensation begins a t 2350 molecule-i at 14.5OC.and at S O O k z at 22.5OC.(1). The existence of these islands was revealed by an application of the Frumkin method for the determination of contact potentials to a two-dimensional survey of the surface of the film in the region LIG of figure 1 (12). Over islands of the phase L1 of myristic acid the surface potential was 150 millivolts, while over por0

24

-

?2 -

I6 -

20

p-

8r

1412-

Y

2 10(L

3 8642-

FIG.3. Pressure-area relations for monolayers of pentadecylic acid on a n aqueous subphase a t pH 2. The hydrochloric acid in the subphase is added to prevent the formation of the calcium or other similar salts of the organic acid. The larger open circles represent pressures above which the film is unstable with reference to the third dimension. A comparison with figure 1 indicates what phases and transformations are involved.

tions where such islands were absent or almost absent this potential fell as low as 5 millivolts or even less. IV.

THE TRIPLE POINT:

GAS-LIQUID(L~)-INTERMEDIATE

The phase relations of three-dimensional systems are often exhibited by diagrams in which pressure and temperature are chosen as the independent variables. However, in the two-dimensional systems under consideration

STATES OF MONOLAYERS

25

the triple point lies a t such a low pressure that the values are inaccurate, so the molecular area is chosen to take its place. Figure 4 shows that the triple point G-L1-I for the normal fatty acids of fro? thirteen to sixteen carbon atoms lies a t a molecular area of from 40 to 44 Aa2and at a temperature which increases about 8°C. for each increment of a carbon atom. The temperatures a t the triple point are: 13 C, -6°C.; 14 C, 5°C.; 15 C, 17OC.; and 16 C, 25°C. The boundary between the vapor and the intermediate phases is omitted, except in the case of the CISacid.

FIG.4. Phase diagram showing the regions of stability for the liquid expanded films of the fatty acids Cla to Cis. V. RELATION BETWEEN MOLECULAR DIMENSIONS AND THE PHASE OF A

MONOLAYER

Since the forces between molecules are highly dependent upon the distances between them, and since the structure of a film depends upon both these forces and the dimensions of the space available for the arrangement of the molecules, it is to be expected that certain limits of mean molecular area will be found to be characteristic of certain types or phases of the film. It is obvious that the area is also dependent upon the size of the molecules which constitute the monolayer. It has been found (5, 17) that in crystals of the normal long-chain paraffin acids, or of the paraffins themselves, the area per molecule perpendicular to the direction of the chain is close to 18.5 A.2 Thus it is not to be expected that a normal long-chain acid, alcohol, amine, or any compound with a single long paraffin chain, can be

26


compressed in a monolayer to a molecular area less than this. Since, however, some water is associated with the monolayer, it is not certain, until it is demonstrated experimentally, that the compression can be carried this far. In crystals of the long-chain acids from C ~ to O C26 the molecular areas of the A, B, and C varieties measured in the plane of the molecular layers are close to 19.5, 20.5, and 23.5 A.z These differences in area are known to be associated with different angles of tilt of the molecules, and these angles are 71", 63", and 53' in the case of stearic acid. In a recent comprehensive paper Dervichian (6) has given a simple theory of the phase changes in monolayers, which would make the discussion presented here unnecessary, if we could subscribe to his ideas. His fundamental assumption is that, on the compression of a monolayer, phase changes begin (or end) a t a molecular area which ( I ) has the same magnitude as that found in the three-dimensional crystal, or ( 2 ) corresponds to where 0 is the volume of the an isotropic film (molecular area = molecule). In hypothesis 1 he implicitly assumes that the molecules in condensed films,-liquid or solid,-are tilted, and that a phase transformation always occurs a t just those areas which correspond to a definite tilt found in three-dimensional crystals. Thus with the area of 18.5 A.z and the angle 53" for crystals of the C form of stearic acid, a phase change should occur at 23.2 A . 2 Dervichian assumes that this area should correspond to the condensation from lowpressure vapor to the LZ liquid phase (figure 5). Table 2 gives Dervichian's experimental values for the molecular areas a t certain phase transformations, together with those predicted by his theory as calculated and the by us from his value for the molecular cross section (18.5 angles of tilt listed by him. The agreement between the theoretical and experimental values of 6ervichian appears, upon merely casual consideration, to substantiate his theory, but confidence in the exact numerical coincidence is lessened by the fact that in extremely careful work, designed to determine these areas accurately, Nutting and Harkins (15) failed to confirm such low values and obtained a value 1.2 kZhigher than the theoretical value for the transformation vapor + Lz, while their work, together with that of others in this laboratory, gives for the area at the transformation LZ+ solid a value 0.6 A . 2 higher than the theoretical and 0.9 A.z higher than Dervichian's experimental result. Phase transformation 2 of table 2 is assumed on the baais of scanty evidence, and phase transformation 4 occurs in an extremely unstable region. We therefore reach the conclusion that there is as yet no sufficient evidence that the areas at which monolayer phase changes

16 28'

*

-

Acid

Margaric

Acid

-

Stearic

Acid

A

Arachidic

17

.

18

o

*a

24

Palmitic

o

c

19

y-

x)

Nonadtcanoic

-

Acid

Acid

A r e a p e r molecule, sp

-

A.

FIG.5. Pressure-area relations of liquid condensed (Ls)films of the normal fatty acids. The curves indicate the increased closeness of approach of the hydrocarbon chains as the molecules increase in length, owing to an increase of van der Waals attraction between the chains. The acids which contain an odd number of carbon atoms transform, as the pressure is increased, into solid monolayers, with a greatly lessened compressibility, a t about 24 dynes cm.-' With an even number of carbon atoms the transition L2+ solid occum if the monolayer is compressed with sufficient rapidity to prevent a transition into the third dimension. The compressions represented here were made more slowly. A t 25°C.the acids which contain from seventeen to twenty carbon atoms exhibit the transition vapor + liquid condensed a t a pressure of a small fraction of a dyne, but the CISacid gives the transition lowpressure vapor + intermediate phase, and the latter phase undergoes a transition to the LIstate at about 6 dynes c n - 1

27

28


occur exhibit exact correspondence with areas in the various three-dimensional forms of the crystals of the same substance. Dervichian (7) interprets his theory as predicting no change of area for the transformation vapor -+ LZas the length of the paraffin chain increases, and considers that the experimental work of Nutting and Harkins cannot be accurate enough to prove such a variation. Their values in A.2 for the molecular areas as related to the number of carbon atoms in the molecule of the normal acid are as follows (figure 5): 16 C = 24.88, 17 C = 24.59, 18 C = 24.41, 19 C = 23.99, 20 C = 25.64 or decreases of 0.29, 0.18, 0.42, and 0.35 respectively. The regular sequence of the values, the small probable error of each molecular area, and the great care taken in the work seem to us t o demonstrate that increase in the number of carbon atoms decreases the molecular areaoof the LZphase in equilibrium with its vapor. The decrease is about 0.31 A.2 per carbon atom added. The solid film is already so much compressed that an increase in the number of carbon TABLE 2 Areas for stearic acid at phase transformations DERVICHIAN

PHASE TRANSFORMATION

1. Vapor -+ L 2 .. . .......................... 2. Assumed change of higher order. . . . . . . . 3. Lt solid... . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Collapse of solid. . . . . . . . . . . . . . . . . . . . . -f

,

NUTSINQ A N D EARLINS

Theory

Experiment

EXemaImENT

23.2 20.8 19.6 18.7

23.6 20.5

24.4

19.3

20.2

18.5

atoms should have a much smaller effect. The m$ecular areas of the normal paraffin-chain alcohols vary only from 22.02 A.2 for the tetradecyl to 21.82 for the octadecyl compound in the LZphase a t E (figure 1). Thus the tighter packing in the liquid condensed phase reduces the compression induced by increased chain length to practically zero. Dervichian assumes that the transformation of the C form of crystal to the A form is continuous in the film through the LZand I regions, beginning a t C and ending a t E. He therefore assumes that this liquid film must be crystalline, and therefore that it corresponds to three-dimensional crystals or a mesomorphous liquid. The whole argument for this idea (reference 6, page 945) is based on the practically exact coorrespondence between the transformation areas for the film, 23.5 and 19.5 A.2, and those for the crystal. However, it has been shown above that these areas are not correct, so no evidence remains for the crystalline nature of the L2 film. The best evidence that the liquid condensed phase is a true liquid, and not of a crystalline type, is the fact that the theory of the viscosity of a monolayer in the liquid state, developed by Moore and Eyring (14) as an extension of Eyring’s theory of the liquid state, was found by Boyd and Harkins (4)t o fit exactly the behavior of the normal long-chain paraffin

ST9TES O F MONOLAYERS

29

acids in the L z state. Thus they found the equation derived from theory log 7 = log 70 k*

+

to give the exact behavior of the acids containing from eighteen to twenty carbon atoms,-namely, stearic, nonadecanoic, and arachidic,-whose pressure-area relations are exhibited in figure 5. The theory gives correctly the general order of magnitude of the constant k. While it would be convenient to have a single characteristic term to designate the Lz phase, it would seem premature at present to adopt Dervichian’s term “mesomorphous,” which is synonymous with “liquid crystal,” and thus suggests, what seems not to be a fact, that this is not a true liquid. The very large change in viscosity and the complete change in the viscosity relations, which occur when the L z is changed t o the solid state, is exhibited more clearly in the work of Fourt and Harkins (9), which shows that the increment in log qo is practically constant for each carbon atom added to the chain, but that the viscosityof the solid, which is almost independent of pressure except near the point of transformation, decreases rapidly with the length of the chain. The orientation of the condensed liquid (Lz) film is in the third dimension, and not in the system considered in only two dimensions. To argue, as some, though not Dervichian, have done, that all monolayer phases which exhibit orientation must, perforce, be liquid crystals or mesomorphous, would support the idea that all monolayers of non-symmetrical molecules must be crystalline, since all monolayer phases of polar-nonpolar molecules on an aqueous subphase exhibit some type of orientation. For example, in pentadecylic acid films the preferred orientation for a dilute gaseous monolayer is parallel to the surface (flat), while in the solid phase it is perpendicular to the surface (upright). In the condensed liquid (L2) the same orientation may be considered to be preferred, but the probability of just this orientation is lessened. In the liquid expanded ( L I )monolayer, orientation in the third dimension is still preserved, but deviations from the perpendicular direction are much more probable. Indeed, it seems that relatively few molecules should be upright. That the orientation in this phase may be greatly changed by increase of pressure is shown by the fact that the mean intermolecular distance may in this phase vary from about 20 to about 60 per cent greater2 than in the solid film or in the liquid film at the higher pressures. If this last distance mere to exist between the hydrocarbon chains, the van der Waals forces would no

* In order to exhibit more clearly what would be involved in an actual separation of the molecules of a liquid by 60 per cent, such an expansion may be imagined to occur in three dimensions. This would involve a swelling of the liquid to four times i t s initial volume. I t is obvious that such a great expansion never occurs in a threedimensional liquid. In the monolayer, however, the expansion in two dimensions may be partly or wholly compensated in the third dimension.

30


longer be able to preserve the liquid state of the film, since the intermolecular van der Waals energy would be reduced to one-seventeenth that in the solid state. However, chains tilted in different directions may come in contact with each other over a portion of their length, and thus give sufficient van der Waals interaction to hold the film together, The hypothesis of a uniform tilt in one direction can hardly be carried so far as to be used in the explanation of the existence of the L1 film. This theory would make it necessary to assume a very large change in angle as the L I phase expands. For example, with pentadecylic acid a t 35.2’C. the increase in area from 29 to 48 A.2 would necessitate a decrease in angle from 39.5’ to 22.5’. While a set of dipoles oriented perpendicular to the surface, and all a t at the same level, should exhibit repulsion, various orientations of the dipoles, m d also various dipole levels and degrees of “hydration,” make it possible that the total effective field of the polar groups of the liquid expanded film may be slightly attractive. This picture of the liquid expanded (L1) film is that of a network of loosely packed molecules held together by regions of contact. In the LI state the adlineation is moderately close to perfect; in the L1 state it is far from perfect. Two other possible theories of this state are outlined in section XV. The characteristics of the intermediate film suggest that it may be a mixture of the LI and L2 phases, but with such small scale regions as to constitute a single phase instead of a mixture of phases. This problem will be discussed in more detail after the thermodynamic characteristics of monolayers are presented. VI. CONDENSED MONOLAYERS OF MOLECULES WHICH LIE FLAT ON THE SURFACE OF WATER

Very long paraffin-chain molecules which have polar groups at each eighth or tenth carbon atom are found to lie flat on the surface of water, and are not compressed into the upright position by increase of pressure, though this may happen if the polar groups in the molecule are spaced much more closely. Thus the w-hydroxydecanoic acids of molecular weights from about to 5 x lo* were found to give condensed liquid films whick exhibited a t the maximum compression possible a thickness of about 4.8 A. and a distance between chains of about 4.0 A. At a pressure of 3 dynes cm.+ the mean thickness of the monolayer for nine such compounds was 4.58 & and the mean molecular spacing 4.32A. The pressurearea isotherms have a form which suggests that above a pressure of about 1.5 dynes cm.-l these films are of the liquid condensed (L1) type, while below this pressure they resemble those usually found for the intermediate phase (16).


31

Such films are much more compressible than those of the LZ phase, which exhibit the upright type of orientation; in accord with this, such films withstand only relatively small pressures, since those of the very long molecules collapse a t about 3 dynes cm.-l, and, with the shortest molecules tested, at about 10 dynes cm.-' VII. THERMODYNAMIC RELATIONS OF TWO-DIMENSIONAL PHASES AND THE TRANSITIONS BETWEEN THEM

For the sake of concreteness the thermodynamic relations of monolayers will be considered for the case of pentadecylic acid. If a large area of the intermediate film of this substance is increased in area by 1 cm.2 by spreading over water, the heat content, H, of the system increases by h, = 300 ergs cm.-2, while in the expanded L1 state the increase is only about 60 ergs cm.-2J or one-fifth as large. The increase of heat content on spreading may be calculated by the use of the equation:

Extension of the film may be defined as a process in which its area is increased while the area of the surface of the water is kept constant. S o w the energy of extension of the film (he) is given by

+

(he), = @*), (he)(2) where f designates a water surface covered by a film and w a clean water surface. Since the increase in heat content of a water surface is about 116 ergs a t the ordinary laboratory temperature

(he),

-

+

(h)! 116

(3)

The increase of heat content on spreading of the liquid condensed film is very much smaller than for the L I or intermediate phases, since it is not far from zero, while for the solid film the value of h, , while not as yet determined for the acid, is very high,-namely, 400 ergs cm.+ for solid cetyl alcohol and probably of the order of 200 ergs em.? for pentadecylic acid. The heat of spreading, q., per is

+

94 = ha 7T (3) Thus since 7~ is not very large for either the intermediate or the L1 phase, the values of h, already listed give a good approximation of the heat absorbed when the film expands by spreading over 1 em.* The entropy of spreading is

32

WILLIAM D. HARKINS AND EDWARD BOYD VIII. CLASSIFICATION O F TWO-DIMENSIONAL PHASES ACCORDING TO THERMODYNAMIC CHARACTERISTICS

Classification A of table 1 assumes the existence of five monolayer phases, while classification B assumes only four. If classification A is adopted, then it may be stated that each phase is associated with its own characteristic value of h, (or of h e , q., qe , sa , se). The values have been given in the preceding section. If classification B is adopted, then a single phase (I11 or intermediate) consists of two subdivisions,-a transition region and a liquid condensed region with extremely different properties, as brought out in table 3. TABLE 3 Comparison of some properties of the liquid condensed ( L * ) ,intermediate, and liquid expanded (L,)states of monolayers (Values illustrate. the general order of magnitude only) PROPERTY

1

I

L2

Heat absorbed on spreading, q,, . . . . . . . . . . . . . . . . . . . Compressibility. . . . . . . . . . . . . :.m8 Hysteresis i n . . . . . . . . . . . . . . . Small Relation of viscosity t o film pressure.. . . . . . . . . . . . . . . . . log q = log q o

I

LI

0 to constant a t 300 60 0.3 0.03 Large Small

~

+ kr

Unknown

Unknown

The viscosity relation given above for the liquid condensed film is that developed theoretically on the basis of the assumption that the film is a true liquid. The experimental values exhibit extremely close accord with the theory. IX. TRANSITION OF T H E THIRD ORDER BETWEEN THE LIQUID CONDENSED

(Lz)

PHASE AND T H E INTERMEDIATE PHASE

Table 3 shows that the liquid condensed ( L z ) and the intermediate region differ very greatly indeed in their characteristic properties, yet they should not be classified as different phases unless there is a phase transition between them. Since there is no apparent kink in the T , U isotherms there cannot be a first-order and probably not a second-order transition. Unfortunately it is difficult to prove experimentally the existence of a third-order transition. All that can be done is to make its existence plausible. A first-order phase change exhibits a latent heat, and a secondorder change, according to Ehrenfest @),has no latent heat, but only a sharp change in the specific heat, while in a third-order change there is only a sharp break in the derivative of the specific heat. Another method of distinguishing the order of a change, also due to Ehrenfest, is discussed by Dervichian. In three dimensions a first-order transformation a t constant pressure and temperature involves a discontinuity in the volume and, in two dimensions, in the area. From equa-

33


tion 7, developed below, it is seen that this means a discontinuity in the first derivative of the free energy of the system with respect to the film pressure. A second-order change involves a discontinuity in the second derivative (equations 8a and 8b), but not in the first, while a third-order change involves a discontinuity in the third derivative (equations 9a, 9b, and Sc), but neither in the first nor in the second. The Gibbs free energy (F) is defined as.: F=E-ST-pv-yo d F = dE

- S d T - T d S - pdv - ~ d -p r d u

(1)

- udr

(2)

The differential of the internal energy (dE) is dE=dQ-dW=

so dF

=

-SdT

TdS-pdu+rda

+ vdp - u d r

and (aFiay),,, =

--(r

or, since d?r = -dr (aF/an),,T =

(aF/aT),,, =

-S

0

= -g/T

(azF/aK~),,T= K = - -

a. uan

(aZF/aT2),,, = C,/T (a3~/an3)p,T = (aZu/an2),,,

At the point D (figure 1) there is evidently no discontinuity in the area and probably not in the compressibility, so there is not a change of the first order and probably not of the second order. At pressures above that at D the compressibility is nearly constant, and (au/an),,r is almost exactly constant. As the pressure is decreased the value of (au/an),,, begins to

34


increase a t D, and increases more and more rapidly as the pressure is lowered. While the experimental work is not sufficiently accurate to demonstrate a discontinuity in the second derivative a2a/aa2,the change from a constant to an increasing value of the first derivative suggests the existence of such a discontinuity. The hypothesis that this may be the case is strengthened by the behavior of the energy relations adjacent t o the point D. Thus the normal long-chain paraffin acl’ds exhibit a constant value of h, which is close to zero between E and D, but a t D the increment heat content on spreading begins to increase, until it rises finally to a very high value, 300 ergs cm.-? This shows that between E and D the heat absorbed when the monolayer expands by either spreading or extension is almost constant and small, but that a t D the amount of heat absorbed begins to increase rapidly until it finally reaches a value of about 300 ergs cm.-2 for spreading and 416 ergs cm.-? for extension. In the case of a normal long-paraffin-chain alcohol, whose molecules are more tightly packed in the LP state, the heat absorbed on spreading or extension increases slowly between E and D, but a t D the rate of increase of the heat absorbed changes to a much higher value. Thus the energy relations at D indicate that a change of the second order does not occur here, but they suggest a discontinuity in the derivative of the specific heat and therefore a third-order change. Thus both the compressibility and the energy relations suggest but do not prove that a third-order change3 occurs a t D. When these relations are considered in connection with the very great difference in thermal and mechanical properties between the regions E D and DB, it seems preferable to classify them as different phases, as has been done in classification A of table 1. X. SECOND-ORDER TRANSFORMATION SOLID -+ LIQUID

Equations have been developed for the heat (Qm) absorbed during the increase of area of a mole of film by spreading or eztension.

By the use of this equation it is found that the transformation solid -+ liquid does not involve the absorption of heat, and therefore cannot be a first-order change. However, it does involve a discontinuity in the compressibility, so it is a change of the second order. a A calculation made after the paper was completed shows t h a t a term which has the general characteristics of a specific heat exhibits the behavior characteristic in three dimensions of a change of the second order. However, the accuracy of the d a t a is not sufficient to make this conclusive, so i t seems probable that the conclusion reached above t h a t i t is of the third order is in better agreement with our total knowledge of the behavior of the monolayer a t D (figure 1).

35


Also (a2F/aT2)= cp/T shows a A-point, and the following relations hold: marked change in viscosity; maximum in contact potential. XI. CHANGE FROM TWO- TO THREE-DIMENSIONAL

SYSTEMS:

COLLAPSE

OF FILMS

While three-dimensional systems do not need to be considered as unstable with reference to a fourth dimension, a two-dimensional system becomes metastable with reference to the third dimension if the pressure is increased sufficiently. The pressure of collapse depends upon: ( 1 ) the composition of the film; ( 2 ) the composition of the subphase; (3) the temperature; (4)the state: nature of the phase; and (5) the speed of compression. The normal long-paraffin-chain acids with an even number of carbon atoms form L:!films which collapse, without transformation into the solid state, a t a pressure of about 20 to 22 dynes cm.-l if compressed slowly, but with very rapid compression there is often a change to the solid state and this may withstand pressures slightly over 60 dynes cm.-l (of the order of 250 atm.) before collapse occurs. With an odd number of carbon atoms, or in the case of alcohols with an odd or even number of carbon atoms, the solid state is formed on slow compression. In f i p r e 3 the large open circles a t the higher pressures and a t about 21 to 23 A.2 indicate pressures above which the film is metastable. The stability limit of pressure increases with the temperature. XII. COLLAPSE OF .4 LIQUID EXPANDED

(L1)OR h COMPRESSED VAPOR

FILX

If the pressure of either a liquid L1 or a vapor film becomes sufficiently high, without a great enough decrease of area to give a transition to the intermediate, the liquid LZ, or the solid state, it will collapse without a two-dimensional change of phase. While substances which spread to monolayers have a positive initial spreading coefficient (S = -AFJ on clean water, the final coefficient, with the monolayer present, is always negative, that is, lenses formed from the monolayer by compression will not spread over it (10, 11). The condition of equilibrium of lenses of oil o with the surface of water w covered by a film f,is

e

+

4 Y. COS 1c. (1) where 8, 4, and 1c. are the angles made with a plane parallel to the surface of the water. The initial spreading coefficient is Y/ COS

= Y,,,,~

COS

S = -AF* = yW - Y ~ -, Y~~ (2) The condition for the existence of the monolayer is that this must be positive. The surface tensions are those of the pure liquids. If these become saturated with each other, S gives the final spreading coefficent which is always negative, but small if the initial coefficient is positive, and

36


especially if it is large. Thus, with a monolayer present, e is sufficiently small to make it possible to assume cos 0 = 1, without introducing an error of any importance, so

+

cos 4 - Yo cos Now the film pressure, a, is defined by the equation Y f = Yw..

a = Yw

- Y/

so

(3) (4)

+

a = Y w - y w , . cos 4 - 7. cos (5) Thus if all the three angles 8, 4, and were zero, then the film pressure would be equal to the initial spreading coefficient, or

+

a =

si

(6)

The exact equation is

Under the conditions for the existence of a monolayer all three angles are small, and if it is remembered that an angle must be over 11’ to have its cosine differ from unity by 2 per cent, it is seen that the film pressure must be very close to the initial spreading pressure, or a

=si

Certain additional features of these relations should be considered. The y. involved in the equation for a is that for the organic liquid saturated

with water, while in the equation for S it is that of the pure organic liquid. However, the effect of water on the surface tension of the organic liquids used in the study of monolayers is so slight that this difference invol9es no appreciable error. The lenses first formed when a monolayer collapses are extremely minute, and may be so small that their spreading pressure is somewhat greater than that of larger lenses. This last factor would give an even closer correspondence between a and S. Since the molecular area at which collapse to liquid lenses occuors, a t the minimum temperature of collapse, is of the order of 28 to 32 A.2 for the normal straight-chain compounds, it is obvious that at this or higher temperatures solid films cannot be formed, since they do not exist much aboye a limit, which is for most compounds of this type not far from 21 A.2 per molecule. XIII. COMPRESSED VAPOR FILMS (ADAM’S “VAPOR EXPANDED”) Certain substances, such as the ethyl esters of the long-chain acids, form films which are highly imperfect gases. Even a t low pressures such as 0.2 to 0.4 dyne cm.-l they have molecular areas very much below that

STATES OF MOh'OLAYERS

37

for a perfect gas. For example, a t 15°C. and 0.4 dyne em.-' the molecular area for ethyl pentadecylate is only one-sixth of the area for a perfect gas. These films may either ( I ) undeigo a transition to the intermediate phase (upon reduction of area this may change to an L2 film, and if the pressure of collapse is not too low, into a solid film), or (2) collapse to give liquid lenses. The intermediate $lm of ethyl margarate a t 15°C. (figure 6) changes to the Lz state a t 22.4 A.? and 2.0 dynes cm.-l and this into the solid phase a t 20d7 A.2 and 6.0 dynes cm.+ This particular solid film collapsed a t 20.8A.? and 19.5 dynes cm.? and exhibits a compressibility of 0.002 a t 20.5 k2molecule-' and 14.5 dynes cm.-l The behavior a t 25°C. of the ethyl esters of from sixteen to nineteen carbon atoms per molecule is exhibited in figure 2. The margarate and palmitate transform into the intermediate state, while the pentadecylate and myristate collapse into the third dimension to form liquid lenses. One of the most important characteristics of this type of film is that a t any given pressure the transformation to the intermediate phase occurs a t a much higher area than if the transformation occurs from the liquid (Ll) state. By extrapolation from Adam's values (2) it may be estimated tha: a t 10°C. this transition occurs in an ethyl palmitate film a t about 80A.Z molecule-', or a t four times the area for close-packed upright chains, while for the L1 -+ I transition for the acid the maximum area is not very faroabove half this value. Even a t 80 As2,however, tbe molecules cannot all lie flat, which would require an area of about 115 A.? molecule-1. Thus we may conclude that this transition does not occur a t any area large enough to allow the flat orientation for all the molecules. More remarkable than this is the fact that the vapor state persists (ethyl myristate at 25°C.) down to an area, about 32, which is not much above one-fourth of the area required for the flat orientation. Thus the hydrocarbon chains must be tilted very far from the flat position. At 15°C. films of ethyl myristate and pentadecylate condense directly into the intermediate state from their vapor phases. According to Adam (3) the film of ethyl palmitate which transforms into the intermediate phase on increase of pressure is also in the gaseous state. However, his pressure-area curves a t high areas, while they seem to support this idea, indicate the possibility that the film may be in the liquid (L1) state, and this should be tested. In any event it may be seen ( I ) that the pressure-area relations for the change into the intermediate phase are independent of the state of the phase to be transformed, that is, whether it is liquid or gaseous, and (2) that the spacing of the curves for the intermediate phase is also independent of whether it is formed from a gaseous or a liquid phase.

38


The obvious explanation of this latter relation is that the spacing of the curves for the intermediate state as a function of the number of carbon atoms corresponds to that for a variation of temperature, and that the latter depends upon the value of the increase ( h ) of the Gibbs heat function for the spreading of the film. This has a value of approximately 200 ergs em.-' for the intermediate phase for the esters under consideration, while for the corresponding acids the value is about 300 ergs em.-' The T), heat absorbed, q, is in this case approximately equal to h (q = h since the film pressure (T) is small. It should be noted, however, that the total heat, Qm, absorbed in the intermediate phase in going from D

+

20 18 I Ethyl margarate at 15 IS'C 2 Ethyl palmtale at 1420'C at 1514'C 4 Ethyl myrstratr d I5 IS'C

I6

3 Ethyl pmtadrcylata

14

-

$12

D

$1

0

z8 6 4

2

Area p r d e d c

FIG.6. Phase diagram for the ethyl esters of the normal fatty acids a t 15°C. By compression the intermediate phase is formed directly from low-pressure vapor (0.1 dyne cm.?) in the case of the margarate, and from highly compressed vapor films in all of the other cases. The doubtful case is that of the palmitate, where the film may be liquid expanded, though Adam considers i t as in the vapor state.

to C (figure 1) is, on account of differences in Au? as large for the esters as for the acids. The heat absorbed (q) in the spreading of the compressed vapor films is very small. At 25°C. the vapor film of ethyl pentadecylrtte collapses a t 31.3 A.2 and 16.6 dynes cm.-', and of the myristate a t 32.0w.2 and 17.6 dynes cm.-' Thus a difference of a carbon atom causes only a slight change in the pressure of transformation into the third dimension. XIV. THE COMPRESSIBILITY OF MONOLAYERS

The compressibility of a monolayer is dependent upon the extent to which the film is already contracted by intermolecular forces. The values


39

may be considered for: (1) substances which exhibit upright molecular orientation in the most compressed state of the film, and ( 2 ) substances, such as the polymers of w-hydroxydecanoic acid, whose molecules lie flat in either dilute or compressed states of the film. Table 3 shows that each kind of phase is characterized by a general order of compressibility. Since the table gives values for relatively few compounds, it is obvious that for other substances or temperatures the limits given below, which refer to this table, may be somewhat extended. Thus the different phases are in general characterized by the values given below : PEASE

I I1 111

IV V

Gas. Perfect gas: TO = nkT Imperfect gas: minimum of same order as for L1 films: e.g., 0.04 for ethyl palmitate a t 25"C., 34 A . 2 and 14 dynes cm.-' Liquid expanded. L1: - K = 2 t o 7 X 10-8 of order of Intermediate, I . Maximum Sormal long-chain acids, 2 to 5 X 10-1 Kormal long-chain esters, 2.2 X 10-1. Minimum same as - I of phase IV = LI Liquid condensed, L p : --I( = 5 t o 10 X 10-8 Solid, S: - X = 7 to 9 X lo-' --I[

The compressibility of the Lz monolayer is of the order of ten times that of the solid, that for L1 is of the order of six times that for Lz,and the maximum for the intermediate phase is of the order of eight times that for L1, while its minimum compressibility is in the limit equal to that of Ls. Since the maximum compressibility of the intermediate phase is somewhat dependent upon the rate of compression, the value for any given substance and temperature is more uncertain than for the other phases. The compressibility of the liquid (L1) phase exhibits an interesting relation in three of the cases thus far investigated. Thus the slope ( L ~ K / ~ u of ) , the compressibility exhibits two values (figure 7). For myrittic acid a t 16.6"C. these are 0.0030 in the low-area roegion (33 to 37.5 A.2) and 0.005 in the high-area region (37.5 to 43.5 A.2), and for pentadecylic acid at 25'C. the values are 0.014 (35 to 39.5 A.z) and 0.003 (39.5 to 46 For tridecylic acid a t 12.5OC. the change of slope occurs a t about 31.5 A.* In the high-area region the compressibility increases much more rapidly as the area approaches closely that a t which vaporization begins. If the above relation could be shown to be general it would indicate a change of a higher order at about the mean molecular area for the LI film. However, for palmitic acid a t 39.85'9. the compressibility isolinear with respect to area from K = 0.016 a t 36 A.2 to K = 0.032 a t 44.5 A.2 and thus

Compressibility

(K)

Tridecylic acid Tridecylic acid Myristic acid Myristic acid Pentadecylic acid Pentadecylic acid Triolein Triolein Ethyl palmitate (gas)

13 14 15 57 18

TABLE 4 of monolayers i n diflerent states

26-31 32-41 31-37.5 37.5-43.5 31.5-39.5 39.5-44 103 126 40

12.5 12.5 16.6 16.6 25.0 25.0 7 7 25

8-16 0-8 8.5

0.019-0.027 0.026-0.06 0.02-0.04 0.04-0.07 0.025-0.035 0.035-0.045 0.03 0.06 0.04

Intermediate state: acids 14 15

~

Myristic acid Pentadecylic acid

1

17.9

Intermediate state: esters Ethyl palmitate Ethyl margarate Ethyl margarate

34.7

20

Pentadecylic acid Palmitic acid Margaric acid Stearic acid Nonadecanoic acid Arachidic acid

21-23.2 21-23.7 21.6-25 21-24 20.4-23.6 20.6-24.2

17 18 18 18 19

Margaric acid Stearic acid Octadecyl alcohol Calcium stearate Ethyl margarate

18 19

__ 15 16 17 18 19

15 15 16 12

0.53 1.2 2.2 ~

RI 1C 7 c 8C 6C

Rz 14C 8C 8C 6C

~

~

~

~

i 30 40 20 0.5 19.5

20.3 20 20.5

j

30 62 55

j ~

18 14 1 15 1 7.5 1 ~

58

40

~

1

27.5 25 25 25 25 25

0.005-0.027 0.008-0.01 0.007-0.0085 0.0063-O.oO95 0.0063-0.0075 0.0065-0.0084

25 22.5 20 22.7 20

0.0008 0.0007 0.0008 0.0020

18 18 18 18

0.013 0.020 0.017 0.026

O.OOO9

41


does not exhibit the behavior described above. It would, therefore, not seem certain that this is a change of a higher order. However, an indication of a change a t just this point has been observed by Dervichian (reference 6, pages 933-4) in the fluidity, apparent dipole moment, and comof triolein a t 3 X 38.3 A.2, pressibility of myristic acid a t 38 to 39 and of tricaproin a t about the same area. He assumes the change to be of the second order in the case of triolein, and of the third order for tricaproin. Dervichian’s general theory is “that areas corresponding to ordinary phase changes are found as points of discontinuity of higher order in those phases which exist a t higher temperatures.” Our observations indicate that these areas are not constant, but shift with the temperature or with the number of carbon atoms in the chain. Dervichian assumes that the discontinuities a t 38 to 39,A.’ occur a t just the area of the triple point, but we obtain the value 43 A.2 as the area a t

Mokcular

31

32

33

34

35

36

37

38

39

Arta

40

i2 41

42

43

44

FIG.7. Variation of compressibility with molecular area

the triple point for the normal acids from tridecylic to palmitic, while, as listedo above, the apparent discontinuities occur a t areas from 31.5 to 39.5 A.2, and seem to exhibit an increasing value as the length of the chain increases. I n section IX it has been assumed, on the basis of the compressibility and the energy relations, that there is a third(possib1y second)-order change a t D (figure 1) from the liquid condensed (L2) to the intermediate film. At this point the compressibility, on going from L2 to I , begins to rise with extreme rapidity and often exhibits the particular type of inflection characteristic of a cubic form. The behavior of the compressibility a t this point is some evidence in favor of a change of a higher order a t this point, as is the behavior of the energy of spreading and extension, as already pointed out. XV. THEORIES O F THE INTERMEDIATE AND EXPANDED LIQUID STATE

It is not the purpose of this paper to present theories of monolayers, but the types of theories which may prove successful may be pointed out.

42


For the intermediate or transition state the two most obvious are: (1) a micelle theory of the intermediate state; (2) a theory that with the expansion of the film in this state rotational (or other) degrees of freedom become active. Theory 1 cannot be the theory of Langmuir, which assumes micelles of constant size, since the theory does not fit the energy relations now known. Thus he obtains a constant energy of expansion (whether of spreading or of extension is not stated) throughout the whole state from D to C (figure l), while actually beyond D the energy rises gradually from a very small to a very large value. An assumption of the existence of micelles of different sizes, with mass law relations between them, may, however, prove more successful. Theory 2 may be developed along the following lines: A hydrocarbon chain may be considered as an ellipsoidal cylinder. The second-order melting of the solid a t E (figure 1) represents the destruction of long-range order. At some point below E energy begins to be taken up in the limited rotation of the molecules. Possibly this may be a t D. At first the rotation is only partial, and energy is taken up slowly. The rotation becomes more complete in certain groups of molecules, that is, the phenomenon is of the cooperative type. At C the rotation is complete and a transfonnation to the liquid expanded (L1) phase occurs, so the heat absorbed on spreading falls to one-fifth of its former value. Such a theory may, when developed, be f Jund to predict that the transformation from intermediate to expanded phase ( I + L,) is a second-order, instead of a diffuse first-order, change. It may also be found that the change from a mesomorphous to an ordinary liquid phase in three dimensions can be explained by the same type of rotational theory as that suggested for this two-dimensional phase change. This would serve as a basis for the designation of the intermediate film as mesomorphous. Useful information could be obtained by determinations of the viscosity of nematic three-dimensional liquids with the groups of (ca. 106) thread-like molecules oriented both along and transverse to the line of flow. This orientation can be maintained by a homogeneous magnetic or electric field. With a magnetic field the long axes arrange themselves parallel to the lines of force. REFERENCES (1) ADAM,K . K . : The Physics and Chemistry of Surfaces.

(2) (3) (4) (5) (6)

The Clarendon Press, Oxford (1930). See reference 1, p. 62. See reference 1, pp. 61-66. BOYD,E . , AND HARKINS, W. D.: J . Am. Chem. SOC.61, 1188 (1939). CLARKSON, C. E., AND MALKIN, T . : J . Chem. SOC. l9S4, 666. DERVICHIAK, D . G.: J. Chem. Phys. 7, 932 (1939).

INTERNAL SURFACE OF CELLULOSIC MATERIALS

43

(7) DERVICHIAN, D . G.:J . Chem. Phys. 8, 347 (1940). P.:Proc. Acad. Sci. Amsterdam 36, 115 (1933). (8) EHRENFEST, (9) FOURT,L., AND HARKINS, W . D . : J . Phys. Chem. 42, 897 (1938). (10) HARKING, W.D.:Colloid Symposium Monograph 6, 20 (1928). W.D . , AND FELDMAN, A.: J. Am. Chem. SOC.44, 2665 (1922). (11) HARKINS, (12) HARKINS, W.D . , A N D FISCHER, E. K . : J . Chem. Phys. 1, 852 (1933). W. D . , AND MYERS,R. J . : Kature 139, 367 (1937). (13) HARKINS, (14) MOORE, W.J., AND EYRING, H . : J. Chem. Phys. 6, 391 (1938). G.C., A N D HARKINS, W. D . : J . .4m. Chem. SOC.61, 1180 (1939). (15) NUTTING, (16) RIES,H.E., HARKINS, W. D . , AND CARMAN, E. F . : J . Am. Chem. Sac. 67, 776 (1935). HARKIXS, W. D . , CARMAN, E. F., AND RIES, H. E . : J. Chem. Phys. 3, 692 (1935). (17) TRILLAT, J. J., AND SOWAKOWSKI, -4.: Ann. phys. [lo]16, 463 (1935).

THE INTERiYAL SURFACE OF CELLULOSIC MATERIALSL ALFRED J. STAMM

AND

MERRILL A. MILLETT

Forest Products Laboratory,P Forest Service, C. S. Department of Agriculture, Madison, W i s eonsi12 Received J u l y 3, 1940 INTRODUCTION

Cellulosic materials have two different types of internal surface: (a) the surface of the microscopically visible structure consisting of either tubular capillaries such as the lumen of fibers in natural organized cellulosic materials, or the interfiber spaces of such processed cellulosic materials as paper, thread, or fabric; and ( b ) the surface of the transient capillaries within the cell walls of all types of cellulosic material that exists only in the presence of a swelling agent. The microscopically visible surface can be estimated from microscopical measurements, from sorption measurements of non-swelling gases or liquids, from the selective adsorption of a solute from a non-swelling solvent, or from the permeability of a mat of fibrous materials to liquids. The surface of the transient capillaries within the cell wall can be estimated from the sorption of polar vapors or liquids, from the selective adsorption of a solute from a swelling solvent, and from a combination of heat of swelling and adhesion tension data. The two different types of surface Presented at the Seventeenth Colloid Symposium, held at Ann Arbor, Michigan, June 6-8, 1940. * Maintained at Madison, Wisconsin, in cooperation with the University of Wisconsin.

The States of Monolayers. - The Journal of Physical Chemistry (ACS

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