J . Phys. Chem. 1989, 93, 6533-6539 has been suggested p r e v i o ~ s l y ' ~as~the ' ~ cause of the rate oscillations. The experiments show that spatial nonuniformities have an important impact on the observed features of local temperatures or overall reaction rate, especially during oscillatory behavior. In some cases propagating reaction fronts lead to simple observed local or overall oscillatory behavior. However, when successive fronts are triggered at different positions, the interaction among the fronts may lead to complex local behavior, such as double peak or more complex oscillations. Thus, dynamic bifurcation between oscillatory behavior of qualitatively different complexity may be due to the spatial nonuniformities and the existence of different triggering positions rather than an exotic reaction mechanism. Hence, spacial care must be exercised in any attempt to determine kinetics using observed bifurcation diagrams or maps of regions having qualitatively different dynamic overall reaction rate. We (18) Lindstrom, T. H.; Tsotsis, T. T. Surf. Sci. 1985, 150, 487.
6533
conjecture that many of the observed chaotic oscillations in the overall reaction rate of catalytic system^'^*^^ are due to the interaction among several reaction fronts and not to an intrinsic chaotic feature of the kinetic mechanism. The experiments indicate that color thermal images are not a sensitive detector of thermal waves, since a uniform heating of a nonuniform surface leads to similar images. The analysis of AT images enables a more accurate detection of thermal fronts and of their main features such as velocity and width. Acknowledgment. We are thankful to the NSF, the Welch Foundation, and the Texas Advance Research Program for support of this research. Registry No. H1,1333-74-0; Ni, 7440-02-0. (19) Razon, L. F.; Chang, S. H.; Schmitz, R. A. Chem. Eng. Sci. 1986, 41, 1561. (20) Sheintuch, M.; Schmidt, J. J. Phys. Chem. 1988, 92, 3404.
Thermogravltatlonal Thermal Diffusion in Electrolyte Solutions. 1. Steady State Frederick H. Horne*vt and Yuan Xu Department of Chemistry, Michigan State University, East Lansing, Michigan 48824 (Received: September 7, 1988)
The full set of nonequilibrium thermodynamic and hydrodyn'amic differential equations is solved for the steady state of thermogravitational thermal diffusion of aqueous solutions of uni-univalent salts, with particular attention to sodium and potassium chloride solutions. Results agree with previous work on nonelectrolyte solutions except that (1) the experimental Soret coefficient in the Hittorf frame differs from the nonelectrolyte coefficient by the thermal expansivity coefficient, and (2) for small temperature gradients or large column length the composition dependence of the overall density (the "forgotten effect") may be important for calculating the value of the Soret coefficient from experimental data.
1. Introduction Interest in thermogravitational thermal diffusion (TGTD) in electrolyte solutions has accelerated in recent years due principally to three developments: (1) increased and more efficient use of TGTD as a means of separating liquid solution components;' (2) improved approaches to the long-sought but so far elusive goal of an explicit usable molecular theory of coupled mass and heat flows in mixtures;2 and (3) the published reports of Gaeta, Perna, Scala, and B e l l ~ c c iwhose ,~ TGTD experiments appear to imply unusual phase-transition behavior in dilute aqueous sodium chloride and potassium chloride solutions. Petit, Renner, and Lin," using a pure thermal diffusion technique, and Naokata and Kimie,4busing a TGTD technique, did not find the behavior suggested by Gaeta et al. Since Gaeta et al. and other TGTD experimentalists have used for their experimental calculations only very approximate equations, developed long ago for gas mixture^,^ and since the Gaeta et al. results are so intriguing, it seemed appropriate to obtain accurate time-dependent equations for TGTD in electrolyte solutions. The results may be readily adapted to nonelectrolyte liquid mixtures and to gas mixtures. In this paper we present the full set of steady-state differential equations and their solutions for the usual TGTD boundary conditions. The results agree in form with those found previously for binary liquid mixtures,6 but the approach to solving the equations here is new. Moreover, specialization to electrolyte solutions leads to a final formula that involves the corrected Soret coefficient u*,' which differs somewhat from the Soret coefficient u of Gaeta et al. 'Present address (to which reprint requests should be sent): College of Science, Oregon State University, Corvallis, OR 9733 1-4608.
The steady-state results serve as both guide and target in obtaining the time-dependent results to be reported in paper 11. Reservoir effects, which can be of great significance in time-dependent experiments, will be treated in paper 111.
2. Transport Equations For a continuous, isotropic, binary mixture at steady state, the equation of conservation of mass is, for each component v.civi = 0
(2.1 )
where ci is the molar density of component i and vi is its local velocity. The barycentric velocity v is the mass-fraction sum of the component velocities (1) Navarro, J. L.; Nadariaga, J. A.; Saviron, J. M. Phys. SOC.Jpn. 1983, 52, 478. (2) (a) Wolynes, P. G. Annu. Reu. Phys. Chem. 1980, 31, 345. (b) Kahana, P. y.;Lin, J. L. J. Chem. Phys. 1982, 74, 2995. (c) Mauzerall, D.; Ballard, S. G. Annu. Rev. Phys. Chem. 1982, 33, 377. (d) Calef, D. F.; Deutch, J. M. Annu. Rev. Phys. Chem. 1983,34,493. ( e ) Fries, P. H.; Patey, G. N. J. Chem. Phys. 1984, 80, 6253. (3) Gaeta, F. S.; Perna, G.; Scala, G.; Bellucci, F. J. Phys. Chem. 1982, 86, 2967. (4) (a) Petit, C. J.; Renner, K. E.; Lin, J. L. J. Phys. Chem. 1984,88,2435. (b) Naokata, T.; Kimie, N. Bull. Chem. SOC.Jpn. 1984, 57, 349. (5) (a) Furry, W. H.; Jones, R. C.; Onsager, L. Phys. Reu. 1939,55, 1083. (b) Tyrell, H. J. V. Diffusion and Heat Flow in Liquids; Butterworths: London, 1961. (c) Alexander, H. F. Z . Phys. Chem. 1954, 203, 212. (6) (a) Horne, F. H.; Bearman, R. J. J. Chem. Phys. 1962,37,2842. (b) Horne, F. H.; Bearman, R. J. J. Chem. Phys. 1967,46,4128. (c) Horne, F. H.; Bearman, R. J . J. Chem. Phys. 1968,49, 2457. (7) (a) deGroot, S. R. L Effet Soret; North-Holland, Amsterdam, 1945. (b) Haase, R. Thermodynamics of Irreversible Process; Addison- Wesley: Reading, MA, 1969.
0022-365418912093-6533$01.50/0 0 1989 American Chemical Society
6534
The Journal of Physical Chemistry, Vol. 93, No. 17, 1989 v =
WIVI
+ w2v2
(2.2)
with wi the mass fraction of component i (2.3) wj = XiMi/M = CiMiP where xi is the mole fraction of component i, Mi is its molar mass, M is the mean molar mass M = X I M I ~2M2 (2.4) and p is the density p = M/V (2.5)
Horne and Xu independent Onsager coefficients G2,Q12, and Om, eq 2.13 reduce to the independent equations
+
Nows
where Vis the molar volume of the solution. The equation for conservation of total mass is obtained by combining eq 2.1-2.3, with the familiar result v.pv = 0
(2.6)
where
The equation of conservation of momentum for a Newtonian fluid subject to no external fields except gravity leads to the equation of motion6.8 V p = -pg - V[ ($7
- C$](V.v)]
- pv-Vv
+ 2V.7 sym Vv
(2.7)
where p is the hydrostatic pressure, g represents the gravitational field, 7 is the shear viscosity, and C$ is the bulk viscosity. The most useful form of the equation of energy transport for experimental purposes is6,8 ( c P / V)V.VT - Tav*Vp = ((I + p1):Vv - V-q - jt.VIHl - (M1/M2)H2](2.8) where cp is the molar constant pressure heat capacity, T is the temperature, a is the thermal expansivity a = v-YaV/anp,,, = -P-l(aP/aT)p,x,
(2.9)
and u is the stress tensor u
I +( : 1 1 +
=-p
-7 - C$ (V-v)
1
27 sym Vv
(2.10)
for a Newtonian fluid. The thermal and mass flux terms in eq 2.8 contain the heat flux q and the molar diffusion flux jy relative to the barycentric velocity
i = 1, 2 j: = ci(vi - v) Note that by eq 2.2, 2.3, and 2.1 1
(2.1 1)
+
(2.12) Mlj? Mj! = 0 The last term on the right-hand side of eq 2.8 contains the partial molar enthalpies HI and H2;this term is proportional to the heat of mixing of the s o l u t i ~ n . ~ ~ ~ ~ The Onsager equations that relate the heat and matter fluxes to the gradients of temperature and chemical potential are = QOIVTPI+ 002VTp2
-jy = Q l , V T p l -*B JI
+ Qoov
In T
+ Q12vTc(2 + QloV In T
- Q 2 l v T p l + Q22vTc12 + Q20vIn T
p22
=
[:
(9) = ax2
2
i=l
M
x2
(2.18)
T,P
3. Practical Transport Equations Although eq 2.1, 2.7, 2.8, and 2.19 suffice as the differential equations for steady-state thermogravitational thermal diffusion in a binary fluid system, they are not those used in practice. In this section we first convert to more common transport parameters such as the mutual diffusivity D,the Soret coefficient u, and the thermal conductivity K . In order to identify the Onsager coefficients of eq 2.19 with conventionally tabulated parameters, it is necessary to define precisely the experimental conditions that underlie the various definitions. A particularly important result of this section is the identification of the Soret coefficient u* determined in thermogravitational thermal diffusion experiments on electrolytes. The thermal diffusion factor a2is defineddsby the experimental equation for the steady state of a one-dimensional pure thermal diffusion experiment in the absence of a pressure gradient (3.1) By eq 2.19 and 3.1 =
(Q2O/QI,)[M2/(MX21122)1
(3.2)
Note that a I = -a2 when a , is defined by the equation symmetric to eq 3.1. The Soret coefficient u is simply7 (3.3)
= a2/T
The equilibrium sedimentation coefficient s2 is defined by the experimental equation for the isothermal equilibrium one-dimensional composition gradient due to a pressure gradient
2
Q
~
~
(2.14)
dX2 =-SXX
j= 1
dz
In eq 2.13 Vpi
]
(2.19)
CJ
~ =Mo = ~
v~/.Li=
In
T,P
with linear dependency relations"
~
(-)a
+ a lnf2
and f 2 is the mole fraction based activity coefficient of the solute. In the experimentally measurable properties mole fraction, pressure, and temperature, eq 2.16 become
a2
(2.13)
1
+ SiVT
(2.15)
dP dz
-
(3.4)
whence
where Si is the partial molar entropy of component i. In the (8) Horne, F. H. J . Chem. Phys. 1966, 45, 3069. (9) Ingle, S . E.; Horne, F. H. J . Chem. Phys. 1973, 59, 5882. (10) Rowley, R. L.; Horne, F. H. J . Chem. Phys. 1980, 72, 131. (11) Bartelt, J. L.; Horne, F. H. J . Chem. Phys. 1969, 51, 210.
Again, si = -s2 if s1 is defined symmetrically. Note that s2is not a transport property since sedimentation is an equilibrium rather than a nonequilibrium phenomenon.
The Journal of Physical Chemistry, Vol. 93, No. 17, 1989 6535
Thermal Diffusion in Electrolyte Solutions TABLE I: Onsaaer Coefficients and Practical Transport Coefficients
of the experiment, when no chemical potential gradient has developed, Fourier’s first law is -q = K ~ V T (3.19) and, by the first of eq 2.16 KO
To obtain the relationship between R12and the mutual diffusion coefficient D defined by Fick‘s law, consider the Fickian diffusion flux jf defined relative to the volume velocity vv jF = ci(vi - vv)
vv = CIVlV,
+ c2v2v2
At the steady state of a thermal diffusion experiment, the diffusion flux vanishes and Fourier’s law in the form -q = K,VT (3.21)
(3.6)
-$ = D(dci/dz)
(3.7)
K,
(3.8)
and the definition of D in eq 3.7 is consistent with both eq 3.8 and the general Gibbs-Duhem result for uniform temperature and pressure
+ V2 d ~ 2= 0
(3.9)
The relationship between the Fickian diffusion flux jF and the barycentric molar diffusion flux j; is
2iF = P(v,/Mdj!
(3.10)
where we have used eq 2.1 1, 2.12, and 3.6. To complete the relationship between D and RI2,we need the relationship between dx2 and dc2, which we obtain from x2 = c2V and the chain rule for d Y
+ aVdT-PVdp
With “practical” transport parameters replacing Onsager coefficients, the flux equations are
+ x ~ dp P
+
= [MI/(MV,]D[Vx2 - X ~ X ~ C T V x~xZS~VP] T -9 = Q2*j!
+ K,VT
(CIC2V132 - C2P)VPl (3.24) The Hittorf diffusion flux is defined by j? = c2(v2 - VI)
(3.25)
jF = ( M / x l M M
(3.26)
It is related to j! by (3.12)
For the one-dimensional, isothermal, isobaric Fick’s law experiment, from eq 3.7, 3.9, and 3.11
Thus, eq 3.23 becomes, for the Hittorf flux -*H J2 - D*[VC~ c~v*VT c~S~*VP]
MI VI -j! = -D-(dx,/dz) PVI v2
where
= [Ml/(MV,]D(dx2/dz)
(3.13)
+
D* = D/CIVI,
By eq 3.13 and 2.19
s2* =
D = -Ql2(v~P22)/(xIM,M2)
(3.14)
The heat of transport Qi* is experimentally obtained, in principle, by determining the heat flux due to matter flux under isothermal conditions. Thus,I0,l2for V In T = 0 q = Q.*’. I JI
(3.15)
Q2* = -(M2/M1) (Q02/ 5212)
(3.16)
and by eq 2.16 By eq 2.12
QI*
= -(MI /M2)Q2*
(3.17)
By eq 3.2 and 3.16, Onsager reciprocity implies Q2
*
= -(Mx2~22/ MI)a 2
(3.23)
Table I is a summary of the relationships between the Onsager coefficients and the practical transport parameters. Although the volume frame of reference is the basis of Fick‘s law and is the reference frame of choice for concentrated electrolyte solutions and for nonelectrolyte mixtures, the Hittorf frame, with the solvent velocity as reference velocity, is the better choice for dilute solutions. Moreover, the composition variable usually chosen for dilute solutions is the molar concentration c2 rather than the mole fraction x2. With eq 3.11, the first of eq 3.22 becomes -*B j2 - [MlV/(MYl)]D[Vc2- (clc2Vla- c2a)VT +
(3.11)
where /3 is the isothermal compressibility. Then Vdcz = ( V I / V ) d ~ -2 X ~ L dY T
+ (M2/M1)(~02~20)/~121~~
+(~2/~1)~(~02~20)/(~12)1 = KO + XI-Q[MI/(VWI~QZ*D (3.22)
-j! VljF + VjF = O
= [%IO = KO
By eq 3.6
d V = (V2- VI) dx2
(3.20)
combined with eq 2.16 yields
where ciV, = (x,V,/V, is the volume fraction of component i. Fick‘s first law is the experimental equation for the relationship between the one-dimensional Fickian diffusion flux jF and the concentration gradient in a binary, isothermal, isobaric system
Vi dcl
= Qoo/T
(3.18)
Two thermal conductivity parameters must be distinguished, in principle, in thermal diffusion experiments. At the beginning (1 2 ) Bearman, R. J.; Kirkwood, J. G.; Fixman, M. Aduances in Chemical Physics; Interscience: New York, 1958, Vol. 1, p 1.
(3.27)
u* = C , V I O - a
ClV,S, - p
(3.28)
Note that cIVl is the volume fraction of solvent and is nearly equal to unity. Even for 0.5 M sodium or potassium chloride solutions, however, the difference, 1 - clVl = c2V2,is about 0.02, and for 1.0 M solutions is about 0.04. It is thus imprudent to neglect c2V2 compared to 1 if 1% or better accuracy is desirable. Moreover, it is incorrect to neglect a priori the thermal expansivity term in the relationship between u* and u since experimental values’ of the Soret coefficient for electrolyte solutions are often comparable to a in magnitude.
4. Experimental Assumptions and Conditions A typical thermogravitational thermal diffusion apparatus is shown in Figure 1. For the thermal steady state the constant temperature TH of the inner cylinder, radius rl, is maintained hotter than the constant temperature Tc of the outer cylinder, radius r2 (a cylindrical jacket surrounds the apparatus). To begin the experiment the apparatus is filled with a solution of concentration c; and is allowed to come to isothermal equilibrium, which is also the sedimentation equilibrium of eq 3.4. By eq 2.7 aP
az = -Pg
6536 The Journal of Physical Chemistry, Vol. 93, No. 17, 1989
Horne and Xu TABLE 11: Values of Some Thermodynamic and Transport Properties for 0.5 M Binary Aqueous NaCl and KCI Solutions at 25 "C"
Figure 1. Schematic profile of the apparatus (not to scale). Inner cylinder radius rI is maintained at higher constant temperature TH;outer cylinder at radius r2 is maintained at lower constant temperature Tc. 2bf = a is the annular spacing, L is the height of the column, r3 is the outer radius of each reservoir, and h is the height of each reservoir.
at mechanical e q u i l i b r i ~ mand ' ~ the pressure is constant in both of the other two directions. By eq 3.27, 3.5, and 2.18
a aZ
- =In c2
s2*
[ 2 :,]
=V R T ~
pg
(4.2)
From Table 11, for both NaCl and KCl, [(Vz/M2)- (Vl/Ml)] = -0.7 X lo4 m3 kg-' and s2*pgi= -2 X m-l. Thus, c2 varies by less than 0.002%/m, and for practical purposes (1) the composition is uniform throughout the column at the beginning of the nonequilibrium experiment and (2) pressure gradient contributions to the composition gradient are always negligible. An immediate consequence is that eq 3.27 becomes
-jr = D*(Vc2 - c2a*VT)
(4.3)
The next simplification comes from the so-called incompressibility assumption. This is based on eq 2.6 and the chain rule equation for the pressure dependence of the density of a pure substance VP = PPVP
where we have again used 7'-v = 0. From eq 3.23, with neglect of pressure terms O=
a M I D ac2 az pV, az
ar
(4.6)
where u = u, is the vertical component of v and where we have used the incompressibility requirement and the neglect of all u, terms t o eliminate also the z dependence of u,. Similar considerations, plus suppression of the pressure term and the heat of mixing term in eq 2.8,6,9*'0yield v.q = 0
%) +
u a t 2 (4.10)
aZ
The radial boundary conditions are u ( r l ) = 0 = u(r2)
(4.5)
aP = -pg + -1 -tr a au r dr
1 -r-( a M I D ac2 - c2u* r ar p v ,
T(ri) = T H , T(r2) = Tc
Thus, if @ = 0, then the fluid is incompressible and Vev = 0. We shall assume this henceforth. If, further, we neglect all azimuthal terms when we convert eq 2.7 to cylindrical coordinates, and if we neglect all terms in the radial component u, of the velocity v, then we have
az
1.80 1.46 1.81 1.97 1.02 4.5 2.86 4.48 4.1 5.99 2.47 -2.25 0.90 -0.02 3.5 1.85 3.7 0.02
contribution of the heat of transport term and the difference between K~ and K,. The final two differential equations stem from eq 2.1 in the form
(4.4)
+ v.vp = 0
KCI soh
1.80 5.84 1.81 1.81 1.02 3.9 2.92 4.92 4.03 6.04 2.45 -0.92 0.93 -0.02 0.9 1.47 0.5 0.02
"Solute is component 2, solvent (H20) is component 1. M I , M2 and VI, V, are the molar masses and partial molar volumes of water and the salts. p is the solution density, a the thermal expansivity, fl the isothermal compressibility, cp the heat capacity, K the thermal conductivity, 9 the shear viscosity, and D the diffusion coefficient. bReference 15. CReference16. 'Reference 17. 'Reference 18. /Reference 19. XReference 20. *Reference 21. 'Reference 22. jEstimated on assumption that the product qD is independent of temperature.
The conservation eq 2.6 can be rewritten as pv.v
NaCl s o h
(4.1 1) The last of these comes from the fact that the radial component of the mass flux vanishes at the walls. Additional boundary conditions are required to obtain c2 as a function of both r and z. These, too, stem from conservation requirements. By symmetry, the average value of c2 at the vertical center plane must be the initial concentration. Thus
( c2($)) = 4
(4.12)
with (4.13)
(4.7) In the steady state, Gauss's divergence theorem requires that
or, from eq 3.22
a ai- = 0
-Kr
ar
ar
(4.14) (4.8)
where we neglect, as we have justified p r e v i o ~ s l y , ~both - ~ * 'the ~
or, by eq 4.10 and 4.11 Jlr2r(Df
(13) Bartelt, J. L.; Horne, F. H. Pure Appl. Chem. 1970, 22, 349.
2
- c2u
dr = 0
(4.15)
The Journal of Physical Chemistry, Vol. 93, No. 17, 1989 6537
Thermal Diffusion in Electrolyte Solutions
At the boundaries
where D’ = M I D / p V l
s ( r l ) = -6,
(4.16)
p
+ pT(T-TM) + pC(c2-c!)
(5.2)
Some useful relationships are
The conditions expressed in eq 4.1 1 and 4.15 are used to solve eq 4.8, 4.6, and 4.10 for T ( r ) , u(r), and c2(r.z). In previous work:Jo,14 we have taken account of the nonconstancy of thermodynamic and transport properties using an elaborate perturbation scheme. Table I1 shows values of the properties for the systems under study here. We have found repeatedly that the only property whose variability significantly affects thermal diffusion is the density, through the Navier-Stokes equation (eq 4.6 here). Consequently, in the remainder of this work, we assume that the viscosity, the thermal conductivity, the partial molar volumes, the Soret coefficient, and the thermal conductivity are constants. We also assume constant diffusivity D‘ as defined by eq 4.16. To incorporate the essential driving mechanism of thermogravitational thermal diffusion, namely, temperature gradient induced convention, we write the density as a truncated Taylor’s series in temperature and composition p =
s(r2) = 6 , s(T) = 0
(r2- r l ) = i ( e 6 - e-*) = 2 i sinh 6 = 276 = a (5.3)
( r 2 / r l )=
The equations to be solved become, for the temperature a2T -=o as2 T(-6) = TH, T(6) = Tc
(5.4)
v(-6) = 0 = u ( 6 )
(5.5)
for the velocity
and for the composition
(4.17)
with
and c; the initial uniform concentration of solute. A bar over a property denotes that the property is to be evaluated at TM and c;. The temperature and composition derivatives are
The steady-state temperature equation, eq 5.4, is quickly solved by quadrature and imposition of the boundary conditions to yield T = T M- ( A T / 2 6 ) s (5.7) with AT = TH
- Tc
(5.8)
In order to solve eq 5.5 we must suppress the concentration part until we have obtained a result for c2. With eq 5.7, and suppression of the concentration term, eq 5.5 becomes
a% _ - -P(g/q)pT(AT / 2 6 ) ~ e ~ ~
Finally, with eq 4.1 as the guide, we set
aP - = -pg
(4.20)
az
(5.9)
as2 Quadrature, imposition of the boundary conditions, and linearization of the exponential lead to
These assumptions when combined with eq 4.6,4.8,and 4.10 yield
u = -T2(g/if)PT(AT/26)(sZ- S2)[s + (s2 + S 2 ) ] (5.10) or u=
-(p.g/v)(a2/48)(Ar)[1
- (s/6)21Ks/6) +
+ (s/6)211 (5.11)
6. Composition Substitution of eq 5.7 and 5.10 into eq 5.6, with introduction of a new radial variable x according to
5. Coordinate Transformation: Temperature and Velocity
x = (s/6),
Equations As before: it is useful to convert to the quasi-linear horizontal variable s, defined by
r = ?es, s = In ( r / T ) ,
i
= (r1r2)Il2
”(a’2 + ax
(5.1)
(14) Horne, F. H.; Anderson, T. G. J . Chem. Phys. 1970, 53, 2321. (15) Millero, F. J. J . Phys. Chem. 1970, 74, 356. (16) Timmermans, Physical and Chemical Constants of Binary Systems;
- x2)[x
ac2
1
a2C2
+ 6 ( 1 + x2)J - -a2(1 + 26x) az 4 ax2
(2+€.)= o
Interscience: New York, 1960. (17) Batuecas, T. Rev. Real Acad. Cienc. Exactas, Fis. Natur. Madrid 1967, 61, 563. (1 8 ) Harned, H. S. Physical Chemistry of Electrolyte Solutiom; Reinhold New York, 1958. (19) Simard, M. A.; Fortier, J. L. Can. J . Chem. 1981, 59, 3208. (20) Out, D.J. P.; Los, J. M. J . Solution Chem. 1980, 9(1), 19. (21) Kestin, J.; Sokolov, M.; Wakeham, W. A. J . Phys. Chem. Re$ Data (?2)’Rard, J. A,; Miller, D. G. J . Solution Chem. 1979, 8(10), 701.
(6.1)
=
ax
-e(]
1978. 7. 941. ..
-1 I x I 1
yields
1 1
where
’
If we neglect terms of order 6 and neglect the second z derivative of c2, eq 6.2 become
6538 The Journal of Physical Chemistry, Vol. 93, No. 17, 1989
-(a ax
8 ~ + 2
neglect of terms of order ( a / 1 9compared )~ to the retained term. The results are
= -Ox(] - x2) dC2
az
(z+...)
ax
Horne and Xu
u,, =
=o
*I
Moreover, eq 4.16 becomes
Thus, through terms of order
1
l , c 2 ( x , L / 2 ) dx = 2 4
(6.5)
Now assume that c2 is separable according to
+
c2 = e-Kz[c(: U(x)] + @(z) + R(x) (6.6) where K is a constant and c: is the initial concentration. Substitution of eq 6.6 into eq 6.4 yields
"[E + t(c;+u) dx dx dx
and
A( d~ + dx d x
cs --(25x - 70x3 80 ko = 21 /4
1
+ 21x5)
(6.15)
t
21tz 4.6 -C(z - L/2) - -x(25 40 80 By the second of eq 6.5
- 70x2
+ 21x4) + C (6.16) (6.17)
and then = KO(c;+u)(l - x2)x
c2 = c;
+ c;
I
(
exp -)::2
( 241; -4)) +
- exp --
(6.7)
..)
= -ex(i - x2)
[ ( 2:7) ( 241gt 4)] -
21x4) exp --
dd dz
- exp --
-
;:
-x(25
- 70x2
+ 21x4) (6.18)
or, after expanding all exponential terms and truncating after the first order in t
( g + t R ) *I = O In order for eq 6.8 to hold, (dd/dz) must be constant, or $=A+Bz
(6.9)
where A and B are both constants. Equations 6.8 and 6.9 are satisfied if R = --BO (15x - lox2 60 l [
1
+ 3x5) - !-t(15x2- 5x4 + x6) + 2
C(l - CX) (6.10) where we have neglected terms of second order in t (after expanding e-fX) and where the integration constants B and C are determined by application of eq 6.5. This yields
(
B = -21t c 49
1 +I-)-; ; ;
(6.1 1)
Since (a/@ = 192vMlD/(p2agVla3AT)L- 0.01 for a = 1 mm and AT = 10 K, we may safely neglect the second term in the denominator of eq 6.1 l and use 21t B=-C (6.12) 48 Note that (C/O) = 9 6 ~ * 1 M , D / ( p ~ a g V , aL-~ )0.05 m-l for' u* = 10-3 ~ - 1 . To solve eq 6.7, we suppose that
u = t U G + ml+ 0 ( ~ 3 ) KO = tko
(6.19) This is the steady-state concentration distribution.
+ t2kl + O(t3)
7. Discussion In section 5, we assumed that the concentration dependence of the density is negligible in eq 5.5. This assumption can now be tested because eq 6.19 is the steady-state concentration distribution. By eq 6.19 216 L c2 - c; = -c;40 2
For c: = 0.5 mol d ~ n - L~ ,= 0.1 m, and
4 0 = 0.05 m-',
then (c2
- c!) = 0.006 mol dmT3= 6 mol w3.From eq 4.19 and Table 11, pc = [M2 - (V2/V1)M1]I0.06 kg mol-' and pc(c-c:) I0.36
kg m-3. This is to be compared with pT(T-TM)= - p a ( T-TM) = 1.5 kg m-3 for AT = 10 K. Thus, the concentration term is indeed smaller than the temperature term in the Navier-Stokes equation. For longer tube length or smaller temperature difference, however, neglect of the concentration term may not be justified. Previous work6 established the steady-state formulas for the derivatives (ac2/ax) and (ac2/az) for thermogravitational thermal diffusion. A formula for c2 itself has not previously been reported. Differentiation of our eq 6.19 for c2 yields, for the radial derivative
(6.13)
Then dx
dx
+ c!)
(:+.;) J,'x(l
= koc!x(l - x2)
L-
--c;o*A 32
7 ( 1 - -x2
+ -x4 21 5
)
(7.2)
and for the horizontal derivative
=o
- 70x?
*I
- x2)uo d x = 0
(6.14)
where the last of eq 6.14 is derived from the first of eq 6.5 with
1
+ 21x4)
J . Phys. Chem. 1989, 93, 6539-6550 These results agree in form with the results for nonelectrolytes.6 The important new features for electrolytes are (1) the present approach is considerably improved in simplicity and accuracy over the previous approach; (2) an explicit formula c2 itself has been
6539
obtained here; and (3) the Soret coefficient u* obtained in TGTD experiments on electrolyte solutions differs from the Soret coefficient u obtained in nonelectrolyte mixtures as discussed at the end of section 3.
ESR Study of the Dynamic Molecular Structure of a Reentrant Nematic Liquid Crystalt Akbar Nayeem and Jack H. Freed* Baker Laboratory of Chemistry, Cornell University, Ithaca, New York 14853-1301 (Received: January 26, 1989)
ESR studies of anisotropic ordering and molecular dynamics using a variety of spin probes (PD-Tempone, MOTA, P-probe, and CSL) in a reentrant nematic (RN) liquid crystal mixture, 60CB-80CB, are described. In order to discern possible differences in the molecular behavior of reentrant mesogens, our results are compared with similar studies in those liquid crystals that, in spite of being structurally similar to 60CB-80CB (e.g., 8CB and S2, which exhibit bilayer smectic-A (SA) phases), do not exhibit reentrant behavior. Such comparisons show the following. (i) The SA-RN transition is very similar to the (normal) N-SA transition. (ii) The orientational ordering of P in the SAand RN phases of 60CB-80CB is consistent with more random packing of the chains than is observed in S2. It is suggested that the collective packing of the chains, which enhances the stability of the SAphase in a normal liquid crystal (e.g., S2), is frustrated in a reentrant nematic liquid crystal. (iii) The ordering of CSL, which packs with the cores, increases smoothly across the N-SA-RN transitions, showing that the packing of the aromatic cores is not sensitive to these phase transitions. In general, there are no dramatic changes in the dynamics of the probes upon entering the RN phase of 60CB-80CB, supporting the belief that the effects driving reentrance are very subtle. However, the reorientational dynamics of the P-probe indicate enhanced packing of the chains in the RN phase.
I. Introduction The phenomenon of reentrance in liquid crystals, wherein a particular mesophase is observed to appear whether one heats or cools the liquid crystal in some given phase, has been studied extensively both by experimental as well as theoretical methods.'.2 The most intriguing aspect of this phenomenon is that when the liquid crystal is cooled, an apparently less ordered phase is formed-an observation that is counterintuitive from a thermodynamic viewpoint. The reentrant thermodynamic phases in these systems are very sensitive to details of the molecular structure; typically, liquid crystalline molecules possessing strongly dipolar heads (e.g., -CN or -NO,)are observed to form reentrant nematic or reentrant smectic phase^,^ in some cases with double or even quadruple reentran~e.~Furthermore, reentrant nematic behavior has only been noted in those systems in which the smectic phase is of the bilayer type. On the basis of these general observations, a principal objective in most studies involving reentrant phases has been to deduce mechanisms leading to plausible molecular models for reentrant behavior, and this has been done most commonly in terms of structure-property relationship^^-^ or phenomenological theories.'* Detailed studies on the molecular ordering and dynamics in reentrant mesophases, on the other hand, have received less attention.'*12 The importance of these latter studies lies in that they can probe the changes in dynamic molecular structure of the liquid crystals as they undergo mesomorphic changes, from which one can better infer the nature of reentrant behavior. The central purpose of this work is to examine the existing models for reentrant behavior in the light of electron spin relaxation studies of spin probes in a reentrant nematic liquid crystal and thereby suggest criteria for discriminating between different mechanisms of reentrance. A detailed study of critical behavior in the (hexy1oxy)biphenyl-(octy1oxy)biphenyl (60CB-80CB) system has been performed using X-ray techniques with a similar view in mind.13 The emphasis in our work, however, is on the dynamics of solutes of different shapes in the solvent 6 0 C B 'Supported by NSF Grant No. DMR 8901718 and NIH Grant No. GM25862.
0022-3654/89/2093-6539$01 .50/0
SOCB, which forms a reentrant nematic phase. We consider here how ESR line-shape studies using a variety of probes dissolved in 60CB-80CB reflect the dynamic molecular structure of the reentrant nematic phase, how these results may be analyzed and reconciled in terms of known models for reentrant behavior, and what can be said with regard to possible differences and similarities between ordinary nematic and reentrant nematic structures in terms of nematic-smectic-A (N-SA) and smectic-A-reentrant nematic (SA-RN) transitions. Accordingly, we describe here careful line-shape studies of motionally narrowed and slow-motional ESR spectra of three deuterated nitroxide radicals, 2,2,6,6-tetramethyl-4-piperidone N-oxide (PD-Tempone), 4-acetamido-2,2,6,6-tetramethylpiperidine N-oxide (MOTA), and 2,2,6,6-tetramethyl-4-[[(butyloxy)(1) For a detailed bibliography covering the literature on reentrant liquid crystals for 1977-1984, see: Hinov, H. P. Mol. Cryst. Liq. Cryst. 1986, 136, 221. (2) Cladis, P. E. Mol. Cryst. Liq. Cryst. 1988, 165, 85. (3) (a) Cladis, P. E . Phys. Reu. Lett. 1975, 35, 48. (b) Cladis, P. E.; Bogardus, R. E.; Aadsen, D. Phys. Reu. A 1978, 18,2292. (c) Suresh, K. A.; Shashidhar, R.; Heppke, G.; Hopf, R. Mol. Cryst. Liq. Cryst. 1983, 99, 249. (4) (a) Sigaud, G.; Tinh, N . H.; Hardouin, F.; Gasparoux, H. Mol. Cryst. Liq. Cryst. 1981,69, 81. (b) Tinh, N . H.; Destrade, C.; Gasparoux, H. Mol. Cryst. Liq. Cryst. 1982, 72, 247. (c) Hardouin, F.; Levelut, A. M. J . Phys. ( k s Ulis, Fr.) 1980, 41, 41. (d) Goodby, J. W.; Walton, C. R. Mol. Cryst. Liq. Cryst. 1985, 122, 219. (e) Fontes, E.; Heiney, P. A,; Halestine, J. L.; Smith, A. B., 111, J . Phys. ( k s Ulis,Fr.) 1986, 47, 1533. (5) Cladis, P. E. Mol. Cryst. Liq. Cryst. 1981, 67, 177. (6) deJeu, W. H. Solid State Commun. 1982, 41, 529. (7) Longa, L.; deJeu, W. H. Phys. Reu. A 1982, 26, 1632. (8) Prost, J. In Liquid Crysrals in One and Two Dimensions; Helfrich, W., Heppke, G., Eds.; Springer-Verlag: Berlin, 1980; p 125. (9) Pershan, P. S.;Prost, J . J . Phys., Left. 1979, 40, L-27. (10) Luckhurst, G. R.; Smith, K. J.; Timimi, B. A. Mol. Crust. Liq. Cryst. 1980, 56, 315. (1 1) Miyajima, S.;Akaba, K.; Chiba, T. Solid State Commun.1984, 49, 675. (12) (a) Dong, R. Y.; Lewis, J. S.;Tomchuk, E.; Bock, E. Mol. Cryst. Liq. Cryst. 1985, 122, 35. (b) Emsley, J. W.; Luckhurst, G. R.; Parsons, P. J.; Timimi, B. A . Mol. Phys. 1985, 56, 767. (13) (a) Kortan, A. R.; Kanel, H. V.; Birgeneau, R. J.; Litster, J . D. Phys. Reo. Lett. 1981, 47, 1206; ( b ) J. Phys. (Les Ulis, Fr.) 1984, 45, 529.
0 1989 American Chemical Society
