J. Phys. Chem. 1991,95, 4534-4540
4534
provide the macroscopic state of affairs. The free energies of transfer of cholesterol from NaC and NaDC (each at 50 m m ~ l d m - ~to) NaC additive and NaDC additive environments (bile salt and additive at 50 mmoldm-3 each; PEG at 0.9% w/v), respectively, are presented in Table 111. It is found that, with minor exceptions, the transfer free energies relative to NaDC are less positive than NaC. Also, more hydrophilic compounds like glucose, fructose, NaA, and urea interact adversely (positive 6(AGo,) values) toward the transfer of the sterol whereas the amphiphile CTAB provides favorable environment with negative 6(AGo,) values. The process of transfer is also spontaneous in the presence of NaS; salicylates are known as better cholesterol solubilizers.I6 The transfer free energy relative to NaDC is more negative than NaC in the NaC NaDC combination; this shows that, at equal concentration, NaDC is a better cholesterol solubilizer than NaC. The deoxycholate has been reported to form coacervate phasewith cationic surfactant-like dodecyltrimethylammonium bromide (DTAB);47 bile salt like sodium chenodeoxycholate and taurochenodeoxycholate have been found also to form coacervates.@ In solution, the bile salt anion and the CTAB cation undergo ion-pairing effect (coacervation has not been observed), forming nonpolar microscopic regions where the lipid (cholesterol) is molecularly dispersed. As mentioned before, the bile salt-CTAB complex may form unilamellar vesicles to effectively solubilize cholesterol. In support of the present findings, a dissolution study of cholesterol in coacervate phase (which is close to the lipid environment in cell membrane) would be worthwhile. A correlation between the free energy of transfer of cholesterol from NaDC to NaDC + additive environments and from NaC to NaC + additive environments is
+
+
+
(44)Cohen, I.; Vaoiliadcs, T. J . Phys. Chem. 1%1,65, 1774. (45) Barry, E. W.; Grey, G. M.T. J. Pharm. Sci. 1974, 63, 548. (46) Ghwh, S.; Das, A. R.; Moulik, S . P. Colloid Polym. Sci. 1979,257, 645. (47) Barry, B. W.; Grey, G. M. T. J. Colloid Inter/ace Sei. 1975,52,327. (48) Jalan, K. N.; Chakraborty, M. L.;Agarwal, S. K.;Samanta, T.; Mahlanobia. D.; Chttoraj, D. K.;Moulik, S.P. J. Surf Sci. Technol. 1986, 2, 1.
presented in Figure 6B. Except for urea and NaA, a linear correlation is observed. The transfer free energy for NaDC system is half that for NaC system; the former is twice as effective as the latter, and this is independent of the types of additives present. A comparable feature is observed in Figure 6A, where NaDC/ cholesterol ratio in presence of additives is half of NaC/cholesterol ratio. The point referring to CTAB is also only expected as argued above, and it is not experimentally obtained. Formation of turbid colloidal solution of cholesterol can take place in certain surfactant media. In previous solubility studies, we had used cholesterol dispersions in sodium dodecyl sulfate.16 Human gall bladder bile is considered to be a combination of mixed micelles containing dispersed cholesterol!* In the course of our solubility study we have observed stable colloidal dispersion of cholesterol at CTAB/NaC ratios greater than unity. Tween 20 has also shown colloidal dispersion of the sterol at concentration greater than cmc. In view of the deposition of cholesterol on the walls of the artery, its transport in blood, and its dispersed presence in gall bladder bile, the study of the formation and stability of the colloidal state of cholesterol in micellar and mixed micellar environment deserves special attention. Conclusion NaDHC has practically no cholesterol-solubilizingcapacity whereas both NaC and NaDC can effectively solubilize the sterol. CTAB enhances the cholesterol-solubiliingcapacities of both NaC and NaDHC. Alone and mixed with additives, NaDC is twice as effective as NaC. CTAB-NaC mixture and Tween 20 can form stable colloidal dispersions of cholesterol. The hydrophilic additives NaA, glucose, fructose, urea, and PEG have reducing effects on the cholesterol solubility in both aqueous NaC and NaDC media. Acknowledgment. This work was conducted under the Department Scientific Assistance Programme of the University Grants Commission, Government of India. Registry No. CTAB, 57-09-0; NaDC, 302-95-4; NaC, 361-09-1; NaDHC, 145-41-5; NaA, 134-03-2; PEG, 25322-68-3; NaS, 54-21-7; cholesterol, 57-88-5; glucose, 50-99-7; fructose, 57-48-7; urea, 57-13-6.
Endoreverslble Thermodynamlcs and Chemlcal Reactions Alexis De Vos Laboratorium v m r elektronika en meettechniek, Rijksuniversiteit re Gent, Sint Pietersnieuwstraat 41, 8-9000 Gent, Belgium (Received: July 6, 1990)
Endoreversible processes were originally modeled as the following special class of irreversible processes: In endoreversible processes the irreversibilities are all located in the transport of heat from the heat sources to the heat engine and from the heat engine to the heat sinks. The present paper demonstrates that it is advantageous to generalize this definition as follows: In endoreversible processes all irreversibilitiesare located in the rransport of heat and/or matter from the heat and/or matter sources to the engine and from the engine to the heat and/or matter sinks. We demonstrate how this generalization allows us to model a class of chemical reactors in the same way as endoreversible heat engines.
1. Introduction
In 1979, Rubin' defined a class of irreversible heat engines as the "endoreversible engines". The working fluid of such an engine undergoes only reversible transformations. All irreversibilities are restricted to the coupling of the engine to the external world. The simplest endoreversible process is the transformation of heat into work by the engine of Curzon and Ahlhorn.2 The
Curzon and Ahlborn engine is an endoreversible engine avant la lettre, as Curzon and Ahlbom described their model in 1975, Le., 4 years before Rubin introduced endoreversibility. In their model all irreversibilities are restricted to the hear exchange between the engine and the external world. The efficiency of the Curzon-Ahlborn engine is limited by the rate at which heat can be exchanged between the working fluid and the two heat reservoirs.
(1) Rubin, M.Optimal configurations of a claas of irreversible engines. Phys. Reo. A 1979, 19, 1272-1276.
(2) Curzon, F.; Ahlborn, B. Efficiency of a Carnot engine at maximum power output. Am. J . Phys. 1975,13, 22-24.
0022-365419112095-4534302.50/0 0 1991 American Chemical Society
Endoreversible Thermodynamics and Chemical Reactions
'9
g1
fhl
Iu Tz I I 5+2 I I E? I u u Figure 1. Three thermodynamic configurations: (a) with four heat reservoirs; (b) with four energy-and-matter reservoirs; (c) with four matter reservoirs.
The hot and the cold heat reservoirs are connected to a reversible Carnot engine by means of thermal resistances. See Figure la. Entropy is created in those two resistors, and therefore the overall engine (i.e., Carnot engine plus two resistors) is irreversible, more precisely endoreversible. Either the Curzon and Ahlborn or the Rubin definition of endoreversible processes was adopted by Salamon et al.,334by De Vos? and by Gordone6 Eventually it found its way t o thermodynamics Endoreversiblemodels for chemical reactors were introduced by Ondrechen et al.9J0 Such models need both heat flows and particle flows. Irreversibilities happen through both the heat transport and the particle transport. Analogously, De Vos" demonstrated, in the framework of solar energy conversion, that a solar cell cannot be modeled as an endoreversible engine, unless heat reservoirs (i.e., reservoirs supplying heat Q, while keeping their temperature T constant) are generalized into heat-and-matter reservoirs (Le., reservoirs supplying both heat flows Q and matter flows N, while keeping constant both their temperature T and their chemical potential p ) and heat conductors are generalized into heat-and-matter conductors. See Figure 1b. To illustrate the usefulness of the extended definition, the present paper will discuss a model where pure matter reservoirs are in contact with a reversible energy converter through matter fluxresistances. The model is relevant to stationaryand to cyclical chemical reactions. The matter reservoir at high chemical potential pl is the reactant reservoir, the convertor is the reaction
The Journal of Physical Chemistry, Vol. 95, No. 11, 1991 4535 chamber (e.g., a fuel cell), and the matter mervoir at low chemical potential p 2 is the reservoir collecting the reaction product. See Figure IC. To stress the interesting analogy between the Curzon and Ahlbom (CA) heat engine and the present endoreversiblechemical engine, we recall the CA engine in section 2, before presenting the chemical engine in section 5 . 2. The Curzon-Ahlborn Engine
Figure l a shows the following: Two heat reservoirs: one at the high temperature T I and one at the low temperature Tz. Two irreversible components: thermal resistors that limit the heat flows QI and Q2;they cause a temperature drop from TI to an intermediate temperature T3and a temperature drop from an intermediate temperature T4 to T2. A reversible Carnot engine between the intermediate heat reservoir at temperature T3 and the intermediate reservoir at temperature T4. To determine the properties of the structure, we only have to implement the two laws of thermodynamics: Axiom 1 : conservation of energy:
where Q are the heat flows (i.e., heat per unit of time), W the work flows (i.e., work per unit of time), and $$ denotes a summation over an arbitrary closed surface. Axiom 2 conservation of entropy:
where .ffdenotes a summation over a closed surface containing only reversible parts, i.e., a surface enclosing only the Camot part. Because of axiom 1 we have QI-Q2-W=O Because of axiom 2 we have
- Q2/
(1)
(2) = If we introduce the following definition of the conversion efficiency Q l / T3
T4
9:
W = VQI then we obtain from (1) and (2) 9
1 - T4/T3
(3)
which is nothing else but the classical proof of Camot's formula, for a reversible engine working between the temperatures T3and T4. Now we need constitutive laws for the thermal conductors. In accordance with Curzon and Ahlborn, we use linear laws:
QI = gl(Ti - T3)
(4)
= gz(T4 - T2)
(5)
Q2
after Fourier's law (3) Salamon, P.; Berry, R. Thermodynamic length and dissipated svailability. Phys. Rev. I r t r . 1983, 51, 1127-1130. (4) A n d m n , B.; Salamon, P.; Berry, R. Thermodynamics in finite time. Phys. Today 1984, Sept, 62-70. (5) De Vos, A. Reflections on the power delivered by endoreversible engines. J . Phys. D Appl. Phys. 1987, 20, 232-236. (6) Gordon, J. Observations on efficiency of heat engines operating at maximum power. Am. J . Phys. 1990,58, 370-375. (7) Callen, H. Thermodynamics and an introduction to thermostatics,2nd ed.; Wiley: New York, 1985; pp 125-127. ( 8 ) Bejan, A. Advanced engineering thermodynamics;Wiley: New York, 1988; pp 404-414.488-522. (9) Ondrechen, M.; Berry, R.; Andresen, B. Thermodynamics in finite time: a chemically driven engine. J. Chem. Phys. 1980, 72, 5 1 18-5 124. (10) Ondrechen, M.; Andrcsen, B.; Berry, R. Thermodynamics in finite time: processes with temperature-dependent chemical reactions. J . Chem. P h p . 1980, 73.5838-5843. ( 1 1 ) De Vos, A. Is a solar cell an endoreversibleengine? S o h Cells 1991, 31, 181-196.
Q = -XV T Straightforward elimination of T3, T4,and Q2from the set of equations (2)-(5) yields
where
Multiplication by
9
then gives
De Vos
4536 The Journal of Physical Chemistry, Vol. 95, No. 1 1 , 1991
T
The Curzon and Ahlborn result not only holds for linear thermal conductors in combination with a Carnot engine but also turns out to be a very good approximation for linear conductors combined with other cyclic engines, such as Otto, Joule, and Diesel engines.l4-I6 A generalization, involving nonlinear conductors satisfying Ql = gl(Ti”- T3”)and Q2 = g2(T4m- Tzm),was discussed by De VOS.~J’
Q1
3. Generalized Endoreversible Engine In the general structure (seeFigure 1 b) we apply three axioms: Axiom 0: conservation of matter:
tW
where N are the particle flows and $$ denotes a summation over an arbitrary closed surface. Axiom 1 : conservation of energy:
where U are the internal energy flows, Ware the work flows, and $$ denotes again a summation over an arbitrary closed surface. Axiom 2: conservation of entropy: Figure 2. Thermodynamic quantities as a function of the conversion efficiency r ) = 1 - T 4 / T 3of a Curzon-Ahlborn engine: (a) heat consumption Q,;(b) work production W; (c) entropy creation rate
s.
Finally, we construct the following expression for the entropy creation rate: s Q2/T2 - Q I / T I = (QI - W / T 2 - Q i / T i = -Q-i - - -~ Q I QI T2 T2 TI = (tic - V ) Q I / T2 (7) where qc denotes the Carnot efficiency 1 - T T I . Equation 7 was already mentioned by De Vos and Pauwels. Substitution of (6)into (7) yields
where $$ denotes a summation over a surface enclosing the reversible part only. See, e.g., refs 18 and 19. Axiom 2, Le.
u2 - P4N u1 - P3N ---T4
T3
leads to VI - W - P ~ N ---UI - P ~ N 7-4 T3
2(
2313
where N denotes N I = N2. This yields
( T I - Tz - T111)’ TI T2(1 - v )
w=
s=g
Figure 2 shows the quantities Q1(v), W(v), and s(v). We see that under reversible conditions, Le., q = 1 - T4/T3qual to its Carnot value vc = 1 - T2/T I ,both Q1and Ware zero. This reminds us of the general principle that under reversible conditions conversion happens infinitely slowly. We see further that for this same value of 7 the entropy creation rate is both minimum and zero. This reminds us of the fact that in a reversible system entropy is constant, whereas in an irreversible system entropy is increasing. We now remark that the produced work W displays a maximum at an v value between 0 and vc. This value is found by solving dW/dv = 0, yielding immediately
s
7
= 1 - (T2/T1)Il2
(
1 - Z)Ul
+
(
2 P 3 - -)N
If we set W=
+ {N
we obtain a “vectorial efficiency” (v,{), with
v = 1 - -T4 T3
T4 r = -P3 T3
- P4
If not only the engine part between reservoir ( T3,cc3)and reservoir ( T4,h4)is reversible but the whole engine is reversible, Le., when T3 = T I ,p 3 = plrT4 = T2,and p4 = p2, then 7 and {take their reversible or Carnot values, given by
This is the expression that constitutes the Curzon and Ahlborn formula. Under this maximum-power condition, the quantities Q1, W, and take nonzero values:
s
(1 2) De Vos, A.; Pauwels, H.On the thermodynamic limit of photovoltaic solar energy conversion. Appl. Phys. 1981, 25, 119-125. (13) Pauwels, H.; De Vos, A. Determination and thermodynamics of the maximum efficiency photovoltaic device. Proceedings of the 15th I.E.E.E. Photovoltaic Specialists Conference, Orlando, 11-15 May 1981; pp 377-382.
(14) Leff, H.Thermal efficienciesat maximum work output: new results for old heat engines. Am. J . Phys. 1987, 55, 602-610. (1 5 ) Landsberg, P. T.; Leff, H.Thermodynamic cycles with nearly universal maximum-power efficiencies. J . Phys. A: Marh. General 1989. 22, 4019-4026. (16) Gordon, J. Maximum power point characteristics of heat engines as a general thermodynamic problem. Am. J . Phys. 1989, 57, 1136-1142. (17) De Vos, A. Efficiency of some heat engines at maximum-power conditions. Am. J . Phys. 1981, 53, 570-573. (18) Callen, H.Thermodynamics; Wiley: New York, 1960; pp 45-46, 293-296. (19) Landsberg, P.T. Thermodynamics and statistical mechanics;Oxford University Press: Oxford, 1978; pp 77-85.
Endoreversible Thermodynamics and Chemical Reactions
The Journal of Physical Chemistry, Vol. 95, No. 11, 1991 4537 as the two terminals are at different chemical potential p ) . Therefore we have to introduce four heat currents: Q I ,Q2,Q3, and Q4,whereas two energy fluxes, VI and U2,sufficed in section 3. See Figure 3. We have Q3 = Q1 + ( p l N - p3N),so that the amount (pl - p3)Nof chemical energy is degraded to heat during the transport. Analogously Q2 is larger than Q4 by an amount (114 - P2)N. Now axiom 2, i.e.
PO-“
leads to Ql
+ plN - p4N-
W
- QI + PIN
T4
- kN
T3
This yields
If we postulate
W = 7Ql Figure 3. Two thermodynamic descriptions: (a) with energy and particle
+ 9N
we obtain a new “vectorial efficiency” (q,S), with
currents; (b) with heat and particle currents.
In nonreversible conditions, the entropy creation rate is given by
with reversible values T2 9c = PI - P2 TI In nonreversible conditions, we find vc=l--
Simple regrouping of terms leads to
This result constitutes a “vectorial product” generalization of eq
7.
Further calculation of the functions W(q,fland s(q,t) will need the constitutive conduction laws for both U and N. We will not perform here such general calculations. We only want to remind that these constitutive laws have to fulfill Onsager’s principle.
4. Alternative Generalized Model As rightly remarked by Callen,’* it is instructive to recast a model, developed in terms of total energy fluxes U and particle fluxes N, into a form in terms of heat fluxes Q and particle fluxes N . For this purpose we define a heat flux Q as U - pN. The three axioms of section 3 then become as follows: Axiom 0: conservation of matter: ##N=O
Axiom I : conservation of energy: $$Q+$$PN+$$W=O
5. Endoreversible Chemical Reactor We will consider here the simple chemical reaction AI - A2 0 where AI constitutes the reactant and A2 the reaction product. Adaption of the calculation for an arbitrary reaction with arbitrary stoichiometric coefficients vj, i.e., for CVjA,
0
(10)
is straightforward. See the Appendix. Because we assume the whole reactor as isothermal (at temperature T), the three general axioms from section 3 lead now to the following two simplified axioms: Axiom 0: conservation of matter: $$N=O
Axiom 2: conservation of entropy in the reversible parts:
$$QP
analogously to (9). A full description based upon the fluxes Q and N is’more complicated than a description based upon the fluxes U and N . Indeed, as mentioned above, we cannot speak of the heat current Q through a particular conductor g. This is caused by the fact that heat fluxes can have divergence, whereas U and N fluxes are always divergenceless. As for a consequence, the notion “constitutive law” cannot be rigorously defined for Q. We will stop here the discussion of the general models, in order to concentrate upon the purely chemical case.
=0
We have to note immediately that axiom 1 has the following consequence: we cannot speak of the “heat current through a thermal conductor”, as the heat entering the conductor at one end does not equal the heat leaving it at the other terminal (as soon
Axiom 1 : summing the conservation law of energy ($$U + $$W = 0) and -T times the conservation law of entropy in the reversible parts (Le., -$NU + $$pN = 0 ) leads to
It is clear that, because of the isothermal assumption, the model
De Vos
4538 The Journal of Physical Chemistry, Vol. 95, No. 11, 1991
t”
TABLE I: Analogy between a Thermal b g h md a Chemical Reactor
thermal engine Fourier’s law Carnot’s law
engine heat engine chemical reactor pneumatic engine electrical circuit
chemical reactor
Q = -XVT
N = -DVn Q = g(T, - T,) N = h(n, n,) q = 1 - T 4 / T 3 j’ = p3 p4 W = ?Q W = j’N = (VC - tl)Q = (tc - O N
-
-
Fick’s law Gibbs’s law
X
Y
transport law
temp T chcm potential p press. p voltage V
heat flow Q particle flow N volume flow u current I
Fourrier’a law Ficlt’s law
Poiseuille’s law Ohm’s law
in chemistry and electrochemistry: it is the diffusion-limited reaction rate. See, e.g., ref 20. For { -c --OD the reaction rate is limited to hlnl;for It += it is limited to -h2n2. For the reaction happening infinitely slowly (N = 0), we have It = Itc = pi - p2 = Ito - k T log (n1/n2),in accordance with the literature?’ Multiplication of (15) by It then gives
-
%
It-
Figwe 4. Thermodynamicquantities as a function of the extracted molar
work j’= p, - p4 of an endoreversible reactor: (a) matter consumption N; (b) work production W; (c) entropy creation rate $.
of section 4 leads to exactly the same two results. The constitutive laws for the two chemical product conductors will be expressed not in terms of the chemical potentials p but in terms of the more common concentrations n. Both variables can be interchanged, because of the existence of a definite relation between them, i.e., the Nernst equation: P = PO
+ k T log (n/no)
It0
Finally, substitution of (15) into
9 = (Itc - l W / T yields the entropy creation rate:
where PO denotes the chemical potential at normalized concentration n., The transport equations are assumed to be linear in the concentrations
- n3) N = hz(n4 - nz)
N
hl(ni
(11) (12)
after Fick’s diffusion law N = -DVn
Because of axiom 1 we have p4N+ W - p 3 N = 0 (13) If we introduce the following definition of It, the “work production per mole of fuel”: It= W / N then we obtain from (1 3)
It = P3 - P4 = It0 + kTlog
(ndn4)
(14)
where to= h3= - h2. In the above we used the notation It. We could have used equally well 9, as both quantities are equivalent in the isothermal case. Straightforward elimination of n3 and n4from (1 l), (12), and (14) yields
Figure 4 shows the quantities W ( { ) and $(It). We see that the curve W ( { ) displays a maximum for some It value between 0 and Itc However, unlike in the case of the Cunon and Ahlborn engine, in the present case, the maximum-power condition (Le., dW/d{ = 0) leads to a transcendental equation. 6. Conclusion
We have developed an analogy between the heat-diffusionlimited heat engine of Curzon and Ahlborn and the particlediffusion-limitedchemical reactor. Table I shows the analogies used. The resulting curves in Figures 2 and 4 are each other’s analogon. The only difference that occurs is the following: all three curves in Figure 2 show a vertical asymptote at Q = 1, whereas none of the three curves in Figure 4 displays a vertical asymptote. This is caused by the fact that p 3 and p4 can take any (positive or negative) value, whereas T3and T4 can take only positive values. Passing the frontier Q = 1 would mean passing below absolute zero for either T3or T4and would need an infinite supply of work (W -m) and would create an infinite amount of entropy (s
--
+a).
It is clear that heat engines and chemical reactors are not the only special applications of the general endoreversible model of Figure l b and section 3. The reader can easily apply the scheme to other systems (pneumatic, hydraulic, etc.). The classioal paper ~~~
Figure 4a shows the function N( 0.The curve N( fl is well-known
~
~~~
(20) Crow, D.Principles ond applications of electrochemistry; Chapman and Hall: London, 1974; pp 175-197. (21) Bcjan, A.Ahneed engimring thedynamics; Wiley: New York, 1988; pp 368-401.
Endoreversible Thermodynamics and Chemical Reactions
The Journal of Physical Chemistry, Vol. 95, No. 11, 1991 4539 such models is then expressed by
where the summation C is performed over all intensive variables
X,except T.
t
Nc
The above illustrates the power of the generalization of the concept 'endoreversible thermodynamics" from merely "irreversible thermodynamics with all irreversibilities caused by hear transport" toward 'irreversible thermodynamics with all irreversibilities caused by transports". Acknowledgment. Alexis De Vos is research engineer in the framework of Imec V.Z.W. (Leuven). The present research was supported by the Commission of European Communities, in the framework of the "Joule" program.
Appendix In order not to make notations too cumbersome, we rewrite the chemical reaction 10 as aA Figure 5. General chemical reactor.
+ bB -tcC + ...
pP
+ qQ + rR +
..,
The stoichiometricnumbers a, b, c, ... and p, 9, r, ... are assumed to be positive integers, so that A, B, C, ... can be called the reactants and P, Q, R, ... the products. See Figure 5 . Axiom 0 is rewritten as follows:
See, e&, ref 23. For convenience we give this quantity the notation N. Axiom 1 leads to
-phzpnZp
c
-------
I
or
W - (up3, + bppb+ cpk
+ ... - ppdP- 9pq - rp4, - ...) N = 0
Taking the Nernst law into account, this becomes
Figure 6. Reaction rate characteristic A'({). Chemical reaction is A + 38 = P + Q. Transport oonductances have bben chosen such that klfil. < 3hl&b and h e b < Full curve: physically relevant branch. Dotted curves: physically irrelevant branches.
of Tolman and Finea2provides us with a good overview of possible applications. Table I1 summarizes some of these. Each time, we recognize reservoirs with a constant value of some intensive variable X and amductors that transport the associated extensive variable flow Y (Le., extensive variable per unit of time). If an engine model contains reservoirs of type ( T X ) , Le., reservoirs at both constant T and constant X , then conservation of entropy in the reversible part of the model is expressed by
Some engine models might need still more elaborate models, e.&, by considering reservoirsof type such as (T#,p), i.e., resmoirs that are simultaneously at amstant temperature, constant pressure, and constant chemical potential, instead of merely reservoirs of the type (T,p). Conservation of entropy in the reversible part of (22) Tolman, R.; Fine, P. On the irreversible production of entropy. Rev. Mod.Phys. 1948, 20, 51-11.
(23) Callen, H. Thermodynomics;Wiley: New York, 1960, pp 199-203.
J. Phys. Chem. 1991, 95,4540-4551
4540
by an example in Figure 6. Of course only the branch that intersects the abscissa is physically relevant. This intersection, by the way, happens at { = Sc, given by
Substitution of (17) and (18) into (16) yields f=
Jb+ k T X
or
The physically meaningful branch has two horizontal asymptotes: lim N = min(ahl,nl,, bhlflIb,chl$lc, ...) C--m
lim N = -min(ph2g2,,, qh2qn2q,r h g 2 , , ...) (-+m
This can be seen as follows. For f
-
-=, eq 19 becomes
This leads to a polynomial equation in N: a polynomial equation in N with obvious roots ahi,ni,, bhibnib, Chigicr... The smallest of these roots is the physically relevant N(-m). Analogously, f +- yields the polynomial equation +
The degree of this equation equals max(a b + c + ...,p
+
+ q + r + ,,.)
with roots
Solution of the equation gives the desired characteristic N( {). It often happens that the equation yields more than one real solution. This gives rise to several branches N ( f ) ,as illustrated
- p h 2 g p , -qhqnar -rh2,4,,
...
where again the value closest to zero has to be retained.
Splnodal Curve of Some Supercooled Llquids Pablo G. Debnetletti,* V. S. Raghavan, and Steven S. Borick Department of Chemical Engineering, Princeton University, Princeton, New Jersey 08544-5263 (Received: November 15, 1990)
There exist two possible limits to the extent to which a liquid can be supercooled: the K a u p n a ~temperature and the spinodal curve. The virial theorem imposes severe constraints on the type of interactionsthat can give rise to logs of mechanical stability upon supercooling and therefore to a supercooled liquid spinodal. Systemscomposed of particles interacting via pair potentials whose repulsive core has a positive curvature (such as the Lennard-Jones potential) cannot become mechanically unstable upon supercooling. Systems composed of particles interacting via potentials whose repulsive a r e is softened by a curvature change are capable of losing stability upon supercooling, and of contracting when heated isobarically. This is consistent with the idea that loss of stability upon supercooling can only occur for liquids capable of contracting when heated. Microscopically, this occurs via the formation of open structures which can be collapsed into denser arrangements through the input of thermal and mechanical energy. In the quasichemical approximation, a very simple model of a core-softened fluid, the lattice gas with attractive nearest-neighbor and repulsive next-nearest-neighbor interactions, exhibits density anomalies in one, two, and three dimensions, and a reentrant, continuous spinodal bounding the superheated, supercooled, and subtriple liquid states in three dimensions.
Introduction Liquids can be cooled below their freezing temperature without solidifying: they can be supercooled. In an ideally purified liquid devoid of any suspended impurity, small nuclei of solid phase are formed exclusively as a result of spontaneous density fluctuations. When such nuclei reach a critical size they grow, and a solid phase To whom correspondence should be addressed.
0022-365419112095-4540$02.50/0
is formed. The formation of critical nuclei (homogeneous nucleation) is a barrier which must be overcome in order for the phase transition to occur.14 In the absence of dissolved impurities, homogeneous nucleation governs the initial rate a t which the (1) Turnbull, D.; Fwher, J. C . J. Chem. Phys. 1949, 17, 71.
(2) Turnbull, D. J . Phys. Chem. 1962, 66, 609. (3) Walton, A. G. Science 1965, 148, 601. (4) Turnbull, D. Contemp. Phys. 1969, 10, 473.
0 1991 American Chemical Society
