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Theory of the effect of phase transition on liquid surface tension
Theory of the Effect of Phase Transition on Liquid Surface Tension LIENG-HUANG
LEE
Research Laboratories Department, Xerox Corporation, Rochester, New York I4603
Received March 15, 1971; accepted March 26, 1971 A phase transition theory for a two-dimensional system containing a liquid or a liquid-like polymer is presented. The theory states: for a two-dimensional system, the criteria for the free energy and its derivatives at a transition are identical to those for a three-dimensional system, except that some higher derivatives either do not exist or are negligible. The theory predicts that surface tension ~, surface entropy per unit area sL and the latent heat of surface formation per unit area 1~ should iump at T,~, the melting temperature, but not at 7'~ , the glass temperature, when 7'0 is treated as a secondorder transition. However, at Tg the second derivative of surface free energy--surface heat capacity %~--should jump. 1. INTRODUCTION
with a variation 2; in area is
I n a previous s t u d y (1) we developed a semiempirieal equation relating surface tension ~/and glass t e m p e r a t u r e Tg of a p o l y m e r t h r o u g h cohesive energy density (C.E.D.). We have continued to search for a unified t h e o r y and a m e t h o d for the determination of solid surface tension. We describe in this paper a t h e o r y developed for the phase transition for a twodimensional s y s t e m containing a liquid or a liquid-like p o l y m e r which is based on the criteria for a three-dimensional system. T h e t h e o r y is t h e n tested with the use of experim e n t a l results published b y others. We then apply the t h e o r y to calculations of solid surface tensions of b o t h glassy and crystalline polymers. T h e i m p a c t of our t h e o r y on the u n d e r s t a n d i n g of other surface and bulk properties of phase transitions is only briefly mentioned. We expect to discuss this i m p a c t in detail in a later publication.
d W = - 3 ` dE + p dr,
where 3' is the surface tension, p, pressure, and v, volume. If the process is isothermal and reversible, the change in the Helmholtz surface free energy (3) is dA ~, which is equal to - d W , and the change in Gibbs surfaee free energy is : dF ~ = 3" dE + v dp.
(aF~/a~)~,~ =
3',
[31
where T is temperature. Surface tension is, therefore, the surface free energy per unit area.
For changes occurring in a unit area of a surface, a n y change of surface energy e~ and surface e n t h a l p y h ~ can be expressed as: e~ =
(aE~/a~)~,
[4]
and h e = ( 5 H ~ / ~ ) ~ = e~ + pay,
The m a x i m u m work (2), dW, for a s y s t e m C o p y r i g h t @ 1971 b y Academic Press, Inc.
[2]
So, if p is also constant,
II. THERMODYNAMIC CONSIDERATIONS A . Surface Energy and Surface Free Energy.
[1]
[5]
Journal of Colloid and Interface Science, Vol. 37, No. 4, December 1971
653
654
LIENG-HUANG LEE
where E is the internal energy, and H, the enthalpy. Since the change of volume Av of the liquid would vanish owing to the choice of the dividing surface, then e~ is equal to h ~, and the system can be treated as a twodimensional system. An analogous Gibbs-Helmholtz equation for surface energy is obtained from the first and second laws of thermodynamics, i.e., e ~ = .y -- T ( ~ . y / ~ T ) p , ~ ,
[6]
~ ,~ \ \
i
\
O
=
-T(~'y/~T)~.~
\
\,
~ %'%%, I TEMPERATURE, OK
Tc
[7]
-
and the latent heat of formation for a unit area of surface is: l~ =
N
dT
j-
at temperatures well below the critical temperature T~, where the surface entropy per unit area is: g
T d--~
\
= T s ~.
2
E e ¢r
.~
[8] z
B. Temperature-Dependence of S u r f a c e T e n s i o n . Figure 1 is a plot of surface tension vs. temperature (4) determined from Eq. [6].
The surface tension of a liquid is usually determined within a narrow range and then extrapolated in both directions to 0°K and the critical temperature T~. As T approaches 0°K, 7 approaches the surface energy e~ per unit area. Thus, e~ is numerically equal to that surface tension y° which the surface of the liquid would possess were it still liquid at absolute zero in the absence of all thermal motion. In fact, there is only helium I I which can remain as a liquid at O°K. The surface energy e~ per unit area remains relatively constant until a certain temperature below 7'~ is reached and becomes zero at T~. The difference between e~ and -/is the latent heat 1~ of surface formation per unit area, which reaches a maximum before reaching Te. In the case of a polymer, Te is a hypothetical temperature, because thermal instability prevents it from reaching that temperature. This type of single-step linear plot has been used to obtain the ~ of a liquid or a polymer at low temperature, even though a liquid or a polymer may undergo a phase Journal of Colloid and Interface Science,
o z
~:
T x = CR_YST_ALLIZATION TEMPERATURE
o
Tx
" %=
TM
T~
TEMPERATURE,°K
3 ' i
[
SUPERCOOLED
~"
r,L,O,D \
~w
I
~: I O
T9
I TM TEMPERATURE,°K
Tc
FIG. 1. Effect of temperature on surface tension (single-step extrapolation). FIG. 2. Changes of surface tension at melting and crystallizaLion temperatures. Fro. 3. Changes of surface tension above and below glass temperature.
Vol. 37, No. 4, December 1971
PHASE TRANSITION THEORY FOR LIQUID SURFACE TENSION
transition. The extrapolation may be legitimate if the material remains as a liquid throughout the entire range of extrapolation. However, for most polymers, this is not the ease since they usually undergo glass or melting transition above room temperature.
C. Surface Entropy and Bulk Properties. The slope of the linear plot described above (and illustrated in Fig. 2) is - s ~. T o examine the change of ~' with respect to phase transition, we should explore those factors controlling £. Let us first study the relations between s~ and such bulk properties as volume V, density o, and thermal expansion coefficient a (5, 6). According to Sudgen (5), paraehor P defined as:
P = @/'~/(V7 ~ - V~-I),
[9]
is a constant and independent of temperature. Where Vl is the molar volume of the liquid, V~ is the molar volume of the vapor, and n is the Macleod constant. The constant n was determined b y Sudgen to be 4 and found b y Wright (7) to be slightly less than 4 for organic liquids. Roe (8) reported n to be between 3 and 4 for several organic polymers. Differentiation (6) of Eq. [9] gives: - (1/-y) (d-~/dT) -~: (n/Vz) (dV~/dT).
[10]
Substituting a in the right-hand side of Eq. [101, we have: -- (1/~') (d~,/dT)
=
- d l n ' y / d T = ha,
[11]
or s ° is equal to nay. We shall employ Eq. [11] for further derivations for the ease where n is available. III. P H A S E T R A N S I T I O N T H E O R Y FOR TWO DIMENSIONAL SYSTEM CONTAINING A LIQUID
Phase transition criteria in a threedimensional system are well established. The application of these criteria to the study of surface tension-temperature dependence has not been investigated. We shall develop a theory for extension of these criteria to a
655
two-dimensional system by making the following assumptions: 1. All the physical properties of a system in a given state of thermodynamic equilibrium should be independent, of time. And there should be no segregation of thermodynamic functions on the basis of geometrical factors, e.g., surfaee or bulk, in regard to the state of equilibrium. 2. At equilibrium, the criteria applicable to thermodynamic functions of the bulk should be equally applicable to those of the surface, since the only difference between the surface and the bulk is that the surface can be considered to be essentially two-dimensional. 3. Since the nonequilibrium glass transition (9) T, bears many eharaeteristies of a true second-order transition T2 (10), the surface thermodynamic funetions can be treated accordingly. On the basis of these assumptions, we state: For a two-dimensional system, the criteria for the free energy and its derivatives at a transition are identical to those for a three-dimensional system, except that. some higher derivatives either do not exist or are negligible. We shall elaborate our theory for various orders of transition in the following seetions.
A. Changes of Surface Tension during Crystal-Melt Transition. The phase change from crystal to melt (11) or vice versa is a first-order transition. Thus, for the transition at melting temperature T,~, the free energy for a two-dimensional system, i.e., the surface free energy, should be continuous: Z~F~ = O.
[121
Furthermore, the first derivative of surface free energy with respect to area should be discontinuous: a~ ~ 0,
[13]
where ( ~ F ~ / ~ ),,~ = .~.
Let us next consider the magnitude of A~.
Journal of Colloid and Interface Science,
Vol. 37, No. 4, December 1971
656
LIENG-HUAN G LEE
For liquid metals, Skapski (12) theorized that t h e crystal should be wetted by its own melt; thus the contact angle 0 is zero. Hence, by applying Young's equation, he obtained: ~
-
w
=
Wl,
[14]
where %z is the interracial tension; -~,, the solid surface tension, and "~l, the surface tension of the melt. On the basis of phase transition criteria established for a two-dimensional system, we would expect ~, to jump at T,~ and we would assume the magnitude of the jump is equal to the interracial tension (Fig. 2) provided t h a t %z is specified for each crystal face i. Therefore, we have ATi
=
(%l)i
=
(7~)i-
"Yl.
[15]
Furthermore, from Eq. [6], we obtain A~'i = --Ali = = (%~)< for each crystal face. Thus, interracial tension is directly related to the latent heat of surface formation other than the heat of fusion (13).
B. Changes of Surface Entropy during Crystal-Melt Transition. In a two-dimensional system, surface entropy g and the latent heat of surface formation l~ should experience the same change as ~, during the crystal-melt transition. B y phase transition criteria, the first derivative of surface free energy with respect to temperature should be discontinuous, i.e., ~ s ~ ~ 0,
water, bismuth, antimony, and gallium (15). A decrease in thermal expansion coefficient also accompanies the density change (14). These normal variations are observed for liquid-like polymers. Therefore, we could eMculate the change of dT'/dT for liquids as well as polymers from the changes in p or a.
C. Changes of Surface Tension during Crystallization. Liquids (13) or polymers generally crystallize at T~, crystallization temperature, which is somewhat lower than T .... T~ also depends on the degree of supercooling. Since Tx is not an equilibrium temperature, we may be able to obtain %x and "Y:l at Tx from the following equation: ~i x =
(~:)i
-
~?
=
(~5)i
; at T~.
[19]
Note that ~ l is also temperature dependent (16). The overall relations between -y and T at both T~ and T~ are illustrated in Fig. 2.
D. Changes of Surface Thermodynamic Functions at Glass Transition. We have assumed that we may treat the nonequilibrium glass transition To as a second-order transition. According to Ehrenfest's definition (17), the surface free energy and any of the first derivatives of the surface free energy should be continuous at T o : ~ v ~ = o,
[20]
A~, = Ag = Al~ = 0.
[21]
and
[16]
Therefore, at glass temperature Tg, the surface tension, the surface entropy, and the latent heat of surface formation should not g = (~F~/ST),,x = (5"~/ST),,,.. [171 jump. T h a t is: Furthermore, 1~ should also be discontin~'l = 3'o, at To, [22] uous: where % is the surface tension of the glassy AV # o. [:81 solid (or noncrystalline solid) at To. EquaTherefore, at the crystal-melt transition tion [22] does not imply that there are two temperature T~, surface tension, surface coexisting phases at To (Fig. 3), as in the entropy, and the latent heat of surface forma- case of T~. tion should jump. Another characteristic of a second-order Generally, the density (14) of a liquid transition (17) is that any of the second (and jumps higher, or the specific volume falls at higher) derivatives of free energy should be T~. However, there are exceptions, e.g., discontinuous. Since the system that we are and
Journal of Colloid and Interface Science, Vol. 37, No. 4, D e c e m b e r 1971
PHASE TRANSITION THEORY FOR LIQUID SURFACE TENSION concerned with is essentially a two-dimensional system in which the change of volume is negligible, we need not deM with the thermal expansion coeffaeient or the compressibility coefficient of the surface component. Therefore, there is only one obvious discontinuous second derivative of the surface free energy, i.e., surface heat capacity cp¢ per unit area. Thus:
c~~ = " T(~2F~/~T~)r,~ = T(Ss~/~T),,z.
[23]
And in the Ehrenfest sense (17), we have A%~ ~ 0. I n other words, the surface heat capacity should jump at T~. This implies that
s~(T > Tg) ~ s~(T < Tg).
[24]
Therefore, we should expect a change in (5"~/~T) above and below T~ (Fig. 3) similar to that. above and below T~ (Fig. 2). And we have
[(~/~T),.~]~>~ ~
[(~%/~T),,~]~<~o. [251
I t is important to note that in Fig. 3 the surface energy e j at 0°K is smaller than that of the crystal e~ at the same temperature. This is presumably due to the difference in density between crystal and glass at. 0°K. In fact, the above criteria are more suitable for a true second-order transition T2 (10). Since T~ is not an equilibrium temperature and is dependent upon other factors, e.g., the thermal history (18) of the sample, we should also expect this of 7 or s ¢.
E. Changes of Surface Thermodynamic Functions at Secondary Transitions. In addition to T.~ and T , , there are minor orderdisorder transitions (18), such as polymorphism transitions or secondary relaxations in the solid state (or in the liquid state as well), which deserve furUher study. If any of these transitions involves changes in p and a, and the changes are significant enough to affect % we should be able to observe a jump or a kink in the 7 -- T plot. We shall devote some of our future work to these subjects, but we shall not discuss them at this time.
657
IV. DISCUSSION
Test of the Theory. 1. The phase transition in a two-dimensional system exists also in surface pressurearea, 7r - A, relation. Harkins (19) observed both first- and second-order transitions under isothermal conditions in his studies of monolayer film. 2. For the crystal-melt transition, a discontinuity in the -y - T plot for paraffin wax was observed by Greenhill and 5~cDonald (20). The iump in ~.~ which they observed at Tm supports our predictions regarding the first-order transition in a two-dimensional system. 3. The magnitude of the jump at T,~, A ~ , was shown to be (~t)~. Though we do not have any direct experimental evidence for polymers, we can point to established indirect evidence, especially that for liquid metals. Skapski (12) claimed to have obtained surface tensions through his equation close to those obtained otherwise. Another familiar example is water (16), for which % (for ice) at 0°C is approximately 100 dynes/ em. This may also be calculated from "7z (76.5 dynes/era) and %z (23.8 dynes/cm) at 0°C. As a result of this predicted jump in surface tension, all solid surface tensions should be higher than those of the melt at T~. For the same reason, the surface tension 7~ of a crystalline polymer should be higher than the ~,~ of the same polymer in the glassy state. For instance, Sehonhorn and Sharpe (21) anticipated the same for two forms of polypropylene. ~I. The kink at Tq in the 7 -- T plot for a polymer has not been directly observed. However, there is indirect evidence for the kink in a polymer solution. Ferroni (22), using a very sensitive surface tension device, detected that kink in a polystyrene solution near its Tg. This result could also support our prediction regarding the kink in the 7 - T plot at Tg for polymers in the absence of a solvent.
Journal of Cdloid and Interface Science, Vol, 37, No. 4, D e c e m b e r 1971
658
LIENG-HUANG LEE
5. The studies carried out b y Allen et al. (23, 24) on the t e m p e r a t u r e dependence of internal pressure P i (--~ C.E.D. ) of polymers also showed a j u m p in P~ at T(2~) and a continuous change at Tg (23). C.E.D. was originally shown b y Hildebrand and Scott (25) to be related to the ~, of a liquid. Our r e c e n t study (26) concluded t h a t at least 65 % of liquids obeyed their equation. Therefore, Allen's results can indirectly confirm our predictions regarding ~, at T~ and at Tg. V. CONCLUSIONS The phase transition theory for a twodimensional system was developed to predict various changes of surface functions at a transition. The theory takes into account b o t h a liquid and a liquid-like polymer at two m a j o r transitions, Tm and Tg. The theory predicts t h a t surface tension % surface entropy s ~ per unit area, and the latent heat of surface formation 1~ per unit area should j u m p at T ~ , but not at T, -provided t h a t Tg is treated as a second-order transition. However, at Tg the second derivative of surface free energy (surface heat capacity per unit area cp¢) should jump, instead. The theory was tested against results published elsewhere.
4. PADDY, J. F., in E. Matijevic, Ed., "'Surface and Colloid Science," Vol. I, p 39. Wiley, New York, 1968. 5. SUEGEN, S., J. Chem. Soc. 125, 32 (1924). 6. BOUTARIC,A., J. Chem. Phys. 31,621 (1934). 7. WRIGHT,F. J., J. Appl. Chem. 11, 193 (1961). 8. ROE, R. ft., Or. Phys. Chem. 72, 2013 (1968). 9. BOYER, R. F., AND SPENCER, R. S., Or. Appl. Phys. 15, 398 (1944). 10. GIBBS, J. I-I., AND DIMARZIO, E. A . , J . Chem. Phys. 28,373 (1958). 11. GORDON,M., in P. D. Ritchie, Ed., "Physics
of Plastics," p 209. Ileffe, London, 1965. 12. SK.~--PSKI,A. S., Acta Met. 4,576 (1956). 13. TURNBULL,D., in F. Seitz and D. Turnbull, Eds., "Solid State Physics, Vol. 3, p 225. Academic Press, New York, 1956. 14. UBEELOHDE, A. R., "Melting and Crystal Structure." p 297. Clarendon Press, 1965. 15. DROsT-H.~NSEN,W., Intern. Sci. and Technol. p 86, (Oct. 1966). 16. DUFOUR, L., AND DEF.&_Y, R., "Thermodynamics of Clouds." Academic Press, New York, 1963. 17. EHRENFEST, P., Commun. Kamerlingh Onnes Lab. Univ. Leiden 756, 1 (1933). 18. BOYER, R. F., Rubber Chem. Technol. 36, 1303 (1963). 19. HARKINS, W. D., "The Physical Chemistry of Surface Films." Reinhold, New York, 1952. 20. GREENHILL, E. B., AND McDONALD, S. R., Nature 171, 37, Jan. 1953.
REFERENCES 1. LEE, L . H . , J. Appl. Polym. Sci. 12,719 (1968).
21. SCHONHORN, H., AND SHARPE, L. H., P o l y m . Lett. 3, 235 (1965). 22. FERI~ONI, E., Atti. Accad. Naz. Lincei, Rend. Cl. Sci. Fis. Mat. Nat. 26,774 776 (1959). 23. ALLEN, G., SIMS, D., AND WILSON, J. G., Polymer 2,375 (1961). 24. ALLEN, G., AND SIMS, D., Polymer 4, 105
2. HARKINS, W. D., AND ALEXANDER, A. E., in A. Weissberg, Ed., "Technique of Organic
25. HILDEBR:kND, J. H., AND SCOTT, R.,"The Solu-
Chemistry," Part I, Chapter 14, p 759, 3rd ed. Interscience, New York, 1963. 3. GIB~s, J. W., "Scientific Papers." Longman and Green, 1906.
bility of Non-Electrolytes." Reinhold, New York, 1950. 26. LEE, L. H., J. Paint Technol. 42, 545, 365 (1970).
(1963).
Journal of Colloid and Interface Science, Vol. 37, No. 4, December1971
LEE
Research Laboratories Department, Xerox Corporation, Rochester, New York I4603
Received March 15, 1971; accepted March 26, 1971 A phase transition theory for a two-dimensional system containing a liquid or a liquid-like polymer is presented. The theory states: for a two-dimensional system, the criteria for the free energy and its derivatives at a transition are identical to those for a three-dimensional system, except that some higher derivatives either do not exist or are negligible. The theory predicts that surface tension ~, surface entropy per unit area sL and the latent heat of surface formation per unit area 1~ should iump at T,~, the melting temperature, but not at 7'~ , the glass temperature, when 7'0 is treated as a secondorder transition. However, at Tg the second derivative of surface free energy--surface heat capacity %~--should jump. 1. INTRODUCTION
with a variation 2; in area is
I n a previous s t u d y (1) we developed a semiempirieal equation relating surface tension ~/and glass t e m p e r a t u r e Tg of a p o l y m e r t h r o u g h cohesive energy density (C.E.D.). We have continued to search for a unified t h e o r y and a m e t h o d for the determination of solid surface tension. We describe in this paper a t h e o r y developed for the phase transition for a twodimensional s y s t e m containing a liquid or a liquid-like p o l y m e r which is based on the criteria for a three-dimensional system. T h e t h e o r y is t h e n tested with the use of experim e n t a l results published b y others. We then apply the t h e o r y to calculations of solid surface tensions of b o t h glassy and crystalline polymers. T h e i m p a c t of our t h e o r y on the u n d e r s t a n d i n g of other surface and bulk properties of phase transitions is only briefly mentioned. We expect to discuss this i m p a c t in detail in a later publication.
d W = - 3 ` dE + p dr,
where 3' is the surface tension, p, pressure, and v, volume. If the process is isothermal and reversible, the change in the Helmholtz surface free energy (3) is dA ~, which is equal to - d W , and the change in Gibbs surfaee free energy is : dF ~ = 3" dE + v dp.
(aF~/a~)~,~ =
3',
[31
where T is temperature. Surface tension is, therefore, the surface free energy per unit area.
For changes occurring in a unit area of a surface, a n y change of surface energy e~ and surface e n t h a l p y h ~ can be expressed as: e~ =
(aE~/a~)~,
[4]
and h e = ( 5 H ~ / ~ ) ~ = e~ + pay,
The m a x i m u m work (2), dW, for a s y s t e m C o p y r i g h t @ 1971 b y Academic Press, Inc.
[2]
So, if p is also constant,
II. THERMODYNAMIC CONSIDERATIONS A . Surface Energy and Surface Free Energy.
[1]
[5]
Journal of Colloid and Interface Science, Vol. 37, No. 4, December 1971
653
654
LIENG-HUANG LEE
where E is the internal energy, and H, the enthalpy. Since the change of volume Av of the liquid would vanish owing to the choice of the dividing surface, then e~ is equal to h ~, and the system can be treated as a twodimensional system. An analogous Gibbs-Helmholtz equation for surface energy is obtained from the first and second laws of thermodynamics, i.e., e ~ = .y -- T ( ~ . y / ~ T ) p , ~ ,
[6]
~ ,~ \ \
i
\
O
=
-T(~'y/~T)~.~
\
\,
~ %'%%, I TEMPERATURE, OK
Tc
[7]
-
and the latent heat of formation for a unit area of surface is: l~ =
N
dT
j-
at temperatures well below the critical temperature T~, where the surface entropy per unit area is: g
T d--~
\
= T s ~.
2
E e ¢r
.~
[8] z
B. Temperature-Dependence of S u r f a c e T e n s i o n . Figure 1 is a plot of surface tension vs. temperature (4) determined from Eq. [6].
The surface tension of a liquid is usually determined within a narrow range and then extrapolated in both directions to 0°K and the critical temperature T~. As T approaches 0°K, 7 approaches the surface energy e~ per unit area. Thus, e~ is numerically equal to that surface tension y° which the surface of the liquid would possess were it still liquid at absolute zero in the absence of all thermal motion. In fact, there is only helium I I which can remain as a liquid at O°K. The surface energy e~ per unit area remains relatively constant until a certain temperature below 7'~ is reached and becomes zero at T~. The difference between e~ and -/is the latent heat 1~ of surface formation per unit area, which reaches a maximum before reaching Te. In the case of a polymer, Te is a hypothetical temperature, because thermal instability prevents it from reaching that temperature. This type of single-step linear plot has been used to obtain the ~ of a liquid or a polymer at low temperature, even though a liquid or a polymer may undergo a phase Journal of Colloid and Interface Science,
o z
~:
T x = CR_YST_ALLIZATION TEMPERATURE
o
Tx
" %=
TM
T~
TEMPERATURE,°K
3 ' i
[
SUPERCOOLED
~"
r,L,O,D \
~w
I
~: I O
T9
I TM TEMPERATURE,°K
Tc
FIG. 1. Effect of temperature on surface tension (single-step extrapolation). FIG. 2. Changes of surface tension at melting and crystallizaLion temperatures. Fro. 3. Changes of surface tension above and below glass temperature.
Vol. 37, No. 4, December 1971
PHASE TRANSITION THEORY FOR LIQUID SURFACE TENSION
transition. The extrapolation may be legitimate if the material remains as a liquid throughout the entire range of extrapolation. However, for most polymers, this is not the ease since they usually undergo glass or melting transition above room temperature.
C. Surface Entropy and Bulk Properties. The slope of the linear plot described above (and illustrated in Fig. 2) is - s ~. T o examine the change of ~' with respect to phase transition, we should explore those factors controlling £. Let us first study the relations between s~ and such bulk properties as volume V, density o, and thermal expansion coefficient a (5, 6). According to Sudgen (5), paraehor P defined as:
P = @/'~/(V7 ~ - V~-I),
[9]
is a constant and independent of temperature. Where Vl is the molar volume of the liquid, V~ is the molar volume of the vapor, and n is the Macleod constant. The constant n was determined b y Sudgen to be 4 and found b y Wright (7) to be slightly less than 4 for organic liquids. Roe (8) reported n to be between 3 and 4 for several organic polymers. Differentiation (6) of Eq. [9] gives: - (1/-y) (d-~/dT) -~: (n/Vz) (dV~/dT).
[10]
Substituting a in the right-hand side of Eq. [101, we have: -- (1/~') (d~,/dT)
=
- d l n ' y / d T = ha,
[11]
or s ° is equal to nay. We shall employ Eq. [11] for further derivations for the ease where n is available. III. P H A S E T R A N S I T I O N T H E O R Y FOR TWO DIMENSIONAL SYSTEM CONTAINING A LIQUID
Phase transition criteria in a threedimensional system are well established. The application of these criteria to the study of surface tension-temperature dependence has not been investigated. We shall develop a theory for extension of these criteria to a
655
two-dimensional system by making the following assumptions: 1. All the physical properties of a system in a given state of thermodynamic equilibrium should be independent, of time. And there should be no segregation of thermodynamic functions on the basis of geometrical factors, e.g., surfaee or bulk, in regard to the state of equilibrium. 2. At equilibrium, the criteria applicable to thermodynamic functions of the bulk should be equally applicable to those of the surface, since the only difference between the surface and the bulk is that the surface can be considered to be essentially two-dimensional. 3. Since the nonequilibrium glass transition (9) T, bears many eharaeteristies of a true second-order transition T2 (10), the surface thermodynamic funetions can be treated accordingly. On the basis of these assumptions, we state: For a two-dimensional system, the criteria for the free energy and its derivatives at a transition are identical to those for a three-dimensional system, except that. some higher derivatives either do not exist or are negligible. We shall elaborate our theory for various orders of transition in the following seetions.
A. Changes of Surface Tension during Crystal-Melt Transition. The phase change from crystal to melt (11) or vice versa is a first-order transition. Thus, for the transition at melting temperature T,~, the free energy for a two-dimensional system, i.e., the surface free energy, should be continuous: Z~F~ = O.
[121
Furthermore, the first derivative of surface free energy with respect to area should be discontinuous: a~ ~ 0,
[13]
where ( ~ F ~ / ~ ),,~ = .~.
Let us next consider the magnitude of A~.
Journal of Colloid and Interface Science,
Vol. 37, No. 4, December 1971
656
LIENG-HUAN G LEE
For liquid metals, Skapski (12) theorized that t h e crystal should be wetted by its own melt; thus the contact angle 0 is zero. Hence, by applying Young's equation, he obtained: ~
-
w
=
Wl,
[14]
where %z is the interracial tension; -~,, the solid surface tension, and "~l, the surface tension of the melt. On the basis of phase transition criteria established for a two-dimensional system, we would expect ~, to jump at T,~ and we would assume the magnitude of the jump is equal to the interracial tension (Fig. 2) provided t h a t %z is specified for each crystal face i. Therefore, we have ATi
=
(%l)i
=
(7~)i-
"Yl.
[15]
Furthermore, from Eq. [6], we obtain A~'i = --Ali = = (%~)< for each crystal face. Thus, interracial tension is directly related to the latent heat of surface formation other than the heat of fusion (13).
B. Changes of Surface Entropy during Crystal-Melt Transition. In a two-dimensional system, surface entropy g and the latent heat of surface formation l~ should experience the same change as ~, during the crystal-melt transition. B y phase transition criteria, the first derivative of surface free energy with respect to temperature should be discontinuous, i.e., ~ s ~ ~ 0,
water, bismuth, antimony, and gallium (15). A decrease in thermal expansion coefficient also accompanies the density change (14). These normal variations are observed for liquid-like polymers. Therefore, we could eMculate the change of dT'/dT for liquids as well as polymers from the changes in p or a.
C. Changes of Surface Tension during Crystallization. Liquids (13) or polymers generally crystallize at T~, crystallization temperature, which is somewhat lower than T .... T~ also depends on the degree of supercooling. Since Tx is not an equilibrium temperature, we may be able to obtain %x and "Y:l at Tx from the following equation: ~i x =
(~:)i
-
~?
=
(~5)i
; at T~.
[19]
Note that ~ l is also temperature dependent (16). The overall relations between -y and T at both T~ and T~ are illustrated in Fig. 2.
D. Changes of Surface Thermodynamic Functions at Glass Transition. We have assumed that we may treat the nonequilibrium glass transition To as a second-order transition. According to Ehrenfest's definition (17), the surface free energy and any of the first derivatives of the surface free energy should be continuous at T o : ~ v ~ = o,
[20]
A~, = Ag = Al~ = 0.
[21]
and
[16]
Therefore, at glass temperature Tg, the surface tension, the surface entropy, and the latent heat of surface formation should not g = (~F~/ST),,x = (5"~/ST),,,.. [171 jump. T h a t is: Furthermore, 1~ should also be discontin~'l = 3'o, at To, [22] uous: where % is the surface tension of the glassy AV # o. [:81 solid (or noncrystalline solid) at To. EquaTherefore, at the crystal-melt transition tion [22] does not imply that there are two temperature T~, surface tension, surface coexisting phases at To (Fig. 3), as in the entropy, and the latent heat of surface forma- case of T~. tion should jump. Another characteristic of a second-order Generally, the density (14) of a liquid transition (17) is that any of the second (and jumps higher, or the specific volume falls at higher) derivatives of free energy should be T~. However, there are exceptions, e.g., discontinuous. Since the system that we are and
Journal of Colloid and Interface Science, Vol. 37, No. 4, D e c e m b e r 1971
PHASE TRANSITION THEORY FOR LIQUID SURFACE TENSION concerned with is essentially a two-dimensional system in which the change of volume is negligible, we need not deM with the thermal expansion coeffaeient or the compressibility coefficient of the surface component. Therefore, there is only one obvious discontinuous second derivative of the surface free energy, i.e., surface heat capacity cp¢ per unit area. Thus:
c~~ = " T(~2F~/~T~)r,~ = T(Ss~/~T),,z.
[23]
And in the Ehrenfest sense (17), we have A%~ ~ 0. I n other words, the surface heat capacity should jump at T~. This implies that
s~(T > Tg) ~ s~(T < Tg).
[24]
Therefore, we should expect a change in (5"~/~T) above and below T~ (Fig. 3) similar to that. above and below T~ (Fig. 2). And we have
[(~/~T),.~]~>~ ~
[(~%/~T),,~]~<~o. [251
I t is important to note that in Fig. 3 the surface energy e j at 0°K is smaller than that of the crystal e~ at the same temperature. This is presumably due to the difference in density between crystal and glass at. 0°K. In fact, the above criteria are more suitable for a true second-order transition T2 (10). Since T~ is not an equilibrium temperature and is dependent upon other factors, e.g., the thermal history (18) of the sample, we should also expect this of 7 or s ¢.
E. Changes of Surface Thermodynamic Functions at Secondary Transitions. In addition to T.~ and T , , there are minor orderdisorder transitions (18), such as polymorphism transitions or secondary relaxations in the solid state (or in the liquid state as well), which deserve furUher study. If any of these transitions involves changes in p and a, and the changes are significant enough to affect % we should be able to observe a jump or a kink in the 7 -- T plot. We shall devote some of our future work to these subjects, but we shall not discuss them at this time.
657
IV. DISCUSSION
Test of the Theory. 1. The phase transition in a two-dimensional system exists also in surface pressurearea, 7r - A, relation. Harkins (19) observed both first- and second-order transitions under isothermal conditions in his studies of monolayer film. 2. For the crystal-melt transition, a discontinuity in the -y - T plot for paraffin wax was observed by Greenhill and 5~cDonald (20). The iump in ~.~ which they observed at Tm supports our predictions regarding the first-order transition in a two-dimensional system. 3. The magnitude of the jump at T,~, A ~ , was shown to be (~t)~. Though we do not have any direct experimental evidence for polymers, we can point to established indirect evidence, especially that for liquid metals. Skapski (12) claimed to have obtained surface tensions through his equation close to those obtained otherwise. Another familiar example is water (16), for which % (for ice) at 0°C is approximately 100 dynes/ em. This may also be calculated from "7z (76.5 dynes/era) and %z (23.8 dynes/cm) at 0°C. As a result of this predicted jump in surface tension, all solid surface tensions should be higher than those of the melt at T~. For the same reason, the surface tension 7~ of a crystalline polymer should be higher than the ~,~ of the same polymer in the glassy state. For instance, Sehonhorn and Sharpe (21) anticipated the same for two forms of polypropylene. ~I. The kink at Tq in the 7 -- T plot for a polymer has not been directly observed. However, there is indirect evidence for the kink in a polymer solution. Ferroni (22), using a very sensitive surface tension device, detected that kink in a polystyrene solution near its Tg. This result could also support our prediction regarding the kink in the 7 - T plot at Tg for polymers in the absence of a solvent.
Journal of Cdloid and Interface Science, Vol, 37, No. 4, D e c e m b e r 1971
658
LIENG-HUANG LEE
5. The studies carried out b y Allen et al. (23, 24) on the t e m p e r a t u r e dependence of internal pressure P i (--~ C.E.D. ) of polymers also showed a j u m p in P~ at T(2~) and a continuous change at Tg (23). C.E.D. was originally shown b y Hildebrand and Scott (25) to be related to the ~, of a liquid. Our r e c e n t study (26) concluded t h a t at least 65 % of liquids obeyed their equation. Therefore, Allen's results can indirectly confirm our predictions regarding ~, at T~ and at Tg. V. CONCLUSIONS The phase transition theory for a twodimensional system was developed to predict various changes of surface functions at a transition. The theory takes into account b o t h a liquid and a liquid-like polymer at two m a j o r transitions, Tm and Tg. The theory predicts t h a t surface tension % surface entropy s ~ per unit area, and the latent heat of surface formation 1~ per unit area should j u m p at T ~ , but not at T, -provided t h a t Tg is treated as a second-order transition. However, at Tg the second derivative of surface free energy (surface heat capacity per unit area cp¢) should jump, instead. The theory was tested against results published elsewhere.
4. PADDY, J. F., in E. Matijevic, Ed., "'Surface and Colloid Science," Vol. I, p 39. Wiley, New York, 1968. 5. SUEGEN, S., J. Chem. Soc. 125, 32 (1924). 6. BOUTARIC,A., J. Chem. Phys. 31,621 (1934). 7. WRIGHT,F. J., J. Appl. Chem. 11, 193 (1961). 8. ROE, R. ft., Or. Phys. Chem. 72, 2013 (1968). 9. BOYER, R. F., AND SPENCER, R. S., Or. Appl. Phys. 15, 398 (1944). 10. GIBBS, J. I-I., AND DIMARZIO, E. A . , J . Chem. Phys. 28,373 (1958). 11. GORDON,M., in P. D. Ritchie, Ed., "Physics
of Plastics," p 209. Ileffe, London, 1965. 12. SK.~--PSKI,A. S., Acta Met. 4,576 (1956). 13. TURNBULL,D., in F. Seitz and D. Turnbull, Eds., "Solid State Physics, Vol. 3, p 225. Academic Press, New York, 1956. 14. UBEELOHDE, A. R., "Melting and Crystal Structure." p 297. Clarendon Press, 1965. 15. DROsT-H.~NSEN,W., Intern. Sci. and Technol. p 86, (Oct. 1966). 16. DUFOUR, L., AND DEF.&_Y, R., "Thermodynamics of Clouds." Academic Press, New York, 1963. 17. EHRENFEST, P., Commun. Kamerlingh Onnes Lab. Univ. Leiden 756, 1 (1933). 18. BOYER, R. F., Rubber Chem. Technol. 36, 1303 (1963). 19. HARKINS, W. D., "The Physical Chemistry of Surface Films." Reinhold, New York, 1952. 20. GREENHILL, E. B., AND McDONALD, S. R., Nature 171, 37, Jan. 1953.
REFERENCES 1. LEE, L . H . , J. Appl. Polym. Sci. 12,719 (1968).
21. SCHONHORN, H., AND SHARPE, L. H., P o l y m . Lett. 3, 235 (1965). 22. FERI~ONI, E., Atti. Accad. Naz. Lincei, Rend. Cl. Sci. Fis. Mat. Nat. 26,774 776 (1959). 23. ALLEN, G., SIMS, D., AND WILSON, J. G., Polymer 2,375 (1961). 24. ALLEN, G., AND SIMS, D., Polymer 4, 105
2. HARKINS, W. D., AND ALEXANDER, A. E., in A. Weissberg, Ed., "Technique of Organic
25. HILDEBR:kND, J. H., AND SCOTT, R.,"The Solu-
Chemistry," Part I, Chapter 14, p 759, 3rd ed. Interscience, New York, 1963. 3. GIB~s, J. W., "Scientific Papers." Longman and Green, 1906.
bility of Non-Electrolytes." Reinhold, New York, 1950. 26. LEE, L. H., J. Paint Technol. 42, 545, 365 (1970).
(1963).
Journal of Colloid and Interface Science, Vol. 37, No. 4, December1971