Andrew G. Dickson Marine Physical Laboratory - 0902, Scripps Institution of Oceanography, University of California, San Diego, 9500 Gilman Drive,
La Jolla, CA 92093-0902, USA
(Received January 13, 1993; revision accepted June 9, 1993)
Abstract
This paper reviews the thermodynamic basis of two approaches that are used to measure the total hydrogen ion concentration of sea water, potentiometry using a glass electrode and spectroscopy using an indicator dye. Both of these methods depend ultimately on measurements made using the classical hydrogen/silver-silver chloride cell for their calibration and thus provide equivalent pH scales. As a result of recent advances in measurement techniques and calibration, we should expect to see a revival in the popularity of pH measurements and a renewed understanding of the importance of this parameter in interpreting acid-base processes in sea water; particularly those involving the geochemically important carbon dioxide system.
1. Introduction
The pH of sea water was first measured over three-quarters of a century ago (Sorensen and Palitzsch, 1910), and for many years pH was measured as a routine chemical parameter on major oceanographic expeditions such as the Norwegian Deep-Sea Expedition (Bruneau et al., 1953) and as recently as GEOSECS (Takahashi et al., 1970). Since then however, the measurement of oceanic pH has largely fallen from favor and nowadays it is only measured infrequently.
The reasons for this are manifold and reflect problems both with the experimental measure- ment itself and with its interpretation. Never- theless, pH is potentially a very valuable oceanographic parameter. It reflects the thermo- dynamic state of all the various acid-base systems present in sea water, particularly the geochemically important carbon dioxide system and is thus indicative of the processes involved in biological production and respiration. Further-
more, it is possible to make pH measurements with a high precision and accuracy, and at a high sampling rate. These characteristics are essential if we are to understand the dynamic nature of the chemical and biological processes taking place in the ocean.
In the past, oceanic pH measurements have usually been made using a potentiometric tech- nique based upon an operational definition of pH. This approach, which uses the hydrogen ion sensitivity of the glass electrode as the basis for the measurement, is beset with a variety of experimental problems such as electrode drift, susceptibility to electromagnetic interference, and problems with reference electrodes. It is this catalogue of potential problems that is in part responsible for the poor opinion many chemical oceanographers have of pH measure- ments. Indeed, recent assessments of potentiometric pH measurements by Working Group 75 of the Scientific Committee on Oceanic Research (SCOR, 1985) state that the usual reproducibility of ocean pH
Fig. 1. Horizontal profiles of sea surface temperature, pH and chlorophyll-a obtained during the Varifront V expedition of 1983 (Fuhrmann and Zirino, 1988).
measurements is no better than +0.02 pH units.
Nevertheless, at least two groups have put together flow-through electrode systems which allow the precise measurement of pH in pumped sea water (Fuhrmann and Zirino, 1988; Mackey et al., 1989). The results published by these investigators illustrate the potential for pH as a useful oceanographic parameter. Fig. 1 (from the work of Fuhrmann and Zirino, 1988) shows clearly the correlation between pH, chloro- phyll-a, and sea surface temperature, and presents a tantalizing glimpse of the informa- tion potentially available in pH measurements.
The measurement problems inherent in potentiometric measurements of pH are almost completely absent from the spectrophotometric measurement of pH, although the necessary calibration information is still sparse. While the degree of precision which can be sustained in potentiometric pH measurements at sea may be
open to debate, it is clear that the advent of the spectrophotometric measurement of ocean pH (Byrne and Breland, 1989; Clayton and Byrne, 1993) sets the precision of sea water pH measure- ments in a new domain. The measurements of Byrne and coworkers indicate that it is now possible to make pH measurements at sea with a precision significantly better than +0.001 pH units. An example of oceanic pH measurements made with this precision is shown in Fig. 2. Note, that in the deep water the measurements agree to within 0.001 pH units.
At this level of precision and sensitivity, sea water pH has the potential to become a valuable parameter for evaluating the internal consistency of measurements of dissolved inorganic carbon, the fugacity of carbon dioxide in equilibrium with a water sample, /(CO2), and total alkalinity, pH is also indicative of those processes that influence the position of equili- brium in the oceanic carbon dioxide system:
Fig. 2. Vertical profile of pH obtained spectrophoto- metrically using the indicator m-cresol purple (calibrated against "tris" buffers). From a NOAA cruise in 1991 at 41°59.6'N, 151°59.1'W (data provided by Tonya Clayton, pers. commun.).
gas exchange, biological uptake/regeneration, mixing. It is therefore essential to understand the basis on which pH measurements are calibrated and the resultant limitations on their use.
2. pH measurements and the study of acid-base processes
The property of interest in any discussion of acid-base processes in sea water is usually dependent on the ratio:
m(HB) m(H +) 7(H+).7(B) 1 - - X x - -
m(B) m ° 7(Ha) K°(HB)
(1)
where m ° has the usual value of 1 mol/kg-H20, and K°(HB) is the standard equilibrium constant for the dissociation reaction:
Sorensen and Linderstrom-Lang (1924) proposed a potentiometric method which they believed measured hydrogen ion activity directly in solution. As a result it became common practice (see for example Bronsted, 1928) to define a hybrid acidity constant, K'(HB), such that:
m(HB) a(H +) m(B) - K'(HB) (3)
or in logarithmic notation
rm(HB)] _- pK'(HB) - pH (4) log[_ m(B) ]
where the notation pX implies - logi0 X; this exponential notation was first proposed by Sorensen (1909) to whom the letter p had the multilingual possibilities of Potenz, puissance or power.
It was quickly appreciated that such measure- ments of pH can not yield a value of a(H +) but rather provide a more nebulous quantity m(H+)7(?) where 7(?) is a complicated function which depends on the transport numbers of all the ions present in all parts of the cell (Guggenheim, 1930). Nevertheless, this in no way vitiates the practical value of this approach. The ratio m(HB)/m(B) is still well defined and such hybrid or "apparent" constants have been extremely useful in a variety of studies in oceanography since they were first introduced by Buch (1930). The only requirement is that pH and pK'(HB) are defined so that Eq. (4) holds true. I shall thus indicate how this is achieved and clarify the restrictions that this requirement places on pH measure- ments of sea water.
(Note that the charges have been omitted from the chemical symbols for the generic acid and base HB and B, and that the concentration unit moles per kg of water - - molality - - is used throughout this paper.)
pH is unique amongst quantities in that in terms definition:
involving as it does a single ion activity, it is immeasurable (Guggenheim, 1930; Covington et al., 1985). The potentiometric measurement of pH is based conceptually on the use of the cell:
where p° is the standard pressure of I01.325 kPa (I atm). Ideally, for this cell,
E = (RT /F) In [m(H+)x/m(H+)s ] (6)
where m(H+)x and m(H+)s are the concen- trations of hydrogen ion in the test and reference solutions respectively.
Cell (A) can be written more practically as
refe ........... trated I test electrode reversible (B) electrode KCI solution ~ solution to H+(aq)
whcre, typically, the clcctrodc reversible to hydrogen ion is a glass electrode that is assumed to behave in a Nernstian fashion. Thus:
pa(H+) = (E °' + Ej) - E (7) RTln lO/F
where Ej is the (formal) liquid junction potential between the two half cells in (B) and E °' = E ° + (RT/F) In a(C1-)Kcv If this cell is used to measure both the test and reference solutions sequentially, then:
Es - Ex AEj pa(H+)x - pa(H+)s = RTln lO/F + RTInlO/F
(8)
AEj is known as the residual liquid junction potential and is the difference in liquid junction
between the test and reference potential solutions.
Eq. (8) definition
is used as the basis of the operational of pH:
E s - E x pH(X) = pH(S) + (9)
RTlnlO/F)
(IUPAC, 1979; Covington et al., 1985). Cell (B) can thus be used to measure differences in pH by potentiometric means; in effect neglecting the residual liquid junction potential, i.e. assuming that AEj = 0.
Although it has been widely recognized that AEj is rarely equal to zero (see e.g. Guggen- heim, 1936; Culberson, 1981; Brezinski, 1983), it is only possible to measure this term in solutions for which the conventional values of a(H +) are known, i.e. solutions with assigned pH(S) values (see e.g. Bates et al., 1950; Coving- ton et al., 1983); this approach necessarily involves some extra-thermodynamic assumption.
The accepted approach to defining reference values for pH buffers in aqueous solution is based on measurements of the e.m.f, of the cell:
solDi~Uont~ aq .... sbuffer I H2(g,p°)Pt(s) Pt(s) Ag(s) AgCl(s) with added NaCl (c)
The reference value, pH(S), for the buffer with- out the addcd NaCl is calculated from thcsc mcasurcmcnts:
EO_E = lim - - - -
pH m(NaCl)~O R T lnlO/F
+ log m(Cl-) + log 7(C1-)} (10)
E ° is the standard potential of the Ag/AgC1 half cell (see e.g. Dickson, 1987). The extra- thermodynamic convention used (Bates and Guggenheim, 1960) is based on the assumption that the activity coefficient of chloride ion in dilute aqueous solution (I < 0.1 mol/kg-H20 ) can be estimated by means of the equation:
where A is the usual temperature-dependent limiting Debye-Hfickel slope and I is the ionic strength.
This is the procedure that was used to deter- mine the pH in a series of primary calibration standards, the so-called NBS buffers (Bates, 1973). It was also the approach taken by Covington et al. (1983) to assess the magnitude of the error in pH measurements made in very low ionic strength media such as lake water.
Unfortunately, in a high ionic strength medium such as sea water, it is not possible to use this approach to define values of pH(S) that
are compatible with those defined in low ionic strength media, pH values measured using the conventional, operational, approach - - using cell (B), Eq. (9) and NBS buffers - - are thus subject to errors which result from the fact that the term 7(?) is a function not only of the com- position of the test and reference solutions, but also of the physical design of the liquid junction used in cell (B). Hence, such pH measurements are necessarily subject to an uncertainty of the order of 0.01 pH units or more (see Dickson, 1984).
4. The measurement of hydrogen ion concentrations in sea water
It is possible to use an alternate convention to define pH standards directly in terms of their total hydrogen ion concentration in sea water, where the activity coefficients of the various reacting acid-base species are dominated by the presence of the bulk electrolyte--the ionic medium. This approach requires that the com- position of the proposed pH calibration standards closely match that of the test solution so that the residual liquid junction term in Eq. (8) is indeed equal to zero. Such buffers are thus made up in a synthetic sea water medium.
In these circumstances, the activity coefficient of hydrogen ion can also be assumed to be identical in the reference and test solutions and Eq. (8) can be rewritten as:
E s - g x pm* (H +)x - pm* (H +)s - R--Tlnl-~-/F (12)
where the asterisk denotes a to ta l hydrogen ion concentration (see below). This expression can then be used as the basis for the potentiometric measurement of total hydrogen concentration; using cell (B) directly in sea water in a fashion analogous to the conventional operational pH measurement. Successful application of this approach depends on being able to assign accurate values for pm*(H+)s to the standard reference buffers.
In a sea water medium, which contains sulfate ion, it is necessary to account for the reaction:
HS04) ~ H + + SO 2- (13)
when discussing hydrogen ion concentrations. In such cases, the formal simplicity of the various mass action, mass balance and Nernst equations for hydrogen ion are retained by defining a to ta l
concentration. This approach was first proposed by Sillrn (Dyrssen and Sillrn, 1967; Sillrn, 1967) and was used by Hansson (1972). It has been refined by Dickson (1990a) who defined the total hydrogen ion concentration in sea water:
m*(U +) = m(n+)[1 + T m ( S O 2 4 - ) / K m ( H S 0 4 ) ]
(14)
where TIn(SO42-) is the total concentration of sulfate ion in the sea water, and
. f m(H+)m(SO 2-) ) l m Km(HSO4) =m(H+)~0~ m ( n S O a ) m (15)
In neutral solutions, where m(HSO4) is small,
m*(H +) ~ m(H +) + m(HSO4) (16)
Eq. (14) is, however, preferred to Eq. (16) as a definition of total hydrogen ion concentration as m* (H +) is then exactly proportional to m(H +) at all pHs. The formation of hydrogen fluoride has not been included in this definition of total hydrogen ion concentration (cf. Dickson and Millero, 1987); it is, of course, still necessary for this acid-base species to be considered explicitly in any discussion of the acid-base chemistry of natural sea water.
The assignment of values of pm*(H+)s to various buffer solutions is based on measure- ments of the cell:
Pt(s) H2(g, p°) water containing AgCl(s) [ Ag(s) I Pt (s) ( D ) HB and B
The e.m.f, of this cell is given by the expression:
E = E ° - ( R T / F ) I n [ m ( H + ) m ( C 1 - ) / ( m ° ) 2]
The value of m(C1-) is known from the buffer composition, it is thus necessary to estimate values for the composite term:
E ~ - (2RT/F) lnT+ (HC1)
+ (RT/F) ln[1 +x m(SO4_)/K,,,(HS04)]
before we can define values of m* (H +) in a buffer solution based on a synthetic sea water. Dickson (1990a) has reported a set of measurements on the cell
Pt(s) H2(g, i atm) Synthetic sea AgCl(s) ] water containing Aft(s) Pt (s) (E) HC1 I
over a wide range of temperatures and salinities; the e.m.f, of this cell is given by Eq. (17). As the total amount of HC1 in the solution was known in these measure- ments and is equivalent to the mass balance expression,
m(HC1) = m(H +) + m(HSO4) - m(OH-) (19)
Eq. (17) is rewritten as:
E = E °
- (RT/F)ln[m(HC1)m(C1-)72(HCl)/(m°) 2]
+ (RT/F)In [1 + m(SO]-)/Qm(HSO4)] (20)
as m(OH-) can safely be ignored in acidic solutions, and
Qm(HSO4) = m(H+)m(SOZa-)/m(HS04) (21)
One then obtains in the limit as m(HC1) ~ 0,
E m = E °
- (2RT/F) In 7~(HC1)
+ (RT/F)In {1 +Tm(SO2-)/Km(HS04)}
= lim [E+ m(HC1)~0
(RT/F) In {m(HC1)m(C1-)/(m°)2}] (22)
The asterisk denotes that the standard state for hydrogen ion is 1 mol/kg-H20 total hydrogen ion - - m*(H +) - - in the synthetic sea water medium, and 7t, r(HC1) is the trace value of the activity coefficient of HC1 in the solution.
These values of E* (Dickson, 1990a) are highly precise and are believed to be accurate to within +0.1 mV (see Fig. 3); this figure has been confirmed by recent measurements by Campbell et al. (1993-this issue). This is
Fig. 3. E' = E + (RT/F) ln[m(HCl)m(C1-)/(m°) 2] plot ted agains t m(HCI). The different symbols represent results f rom two independen t laborator ies ( K h o o et al., 1977; Dickson, 1990a).
equivalent to an error of about 0.002 pH (or pK) units. A comparison of Eqs. (18) and (22) indi- cates that it is possible to assign values for m*(H+)s in buffer solutions provided that the activity coefficient ratio 7±(HC1)/7~(HCI) is assumed to be unity; 7±(HC1) is the actual activity coefficient in the buffer solution, 7~(HC1) is the trace activity coefficient in pure ionic medium. This implies that the small amounts of buffer substances present in the synthetic sea water have no effect on 7±(HC1). This is probably a reasonable assumption in most cases (see Dickson, 1993).
Professor Bates and coworkers at the Uni- versity of Florida have published a series of papers detailing emf measurements on various nitrogen bases in synthetic sea water. They used cell (D) to study the acid dissociation of the hydrogen forms of: 2-amino-2-hydroxy- methyl-l,3-propanediol ("tris"; Ramette et al., 1977); 2-amino-2-methyl-l,3-propanediol ("bis"; Bates and Calais, 1981); tetrahydro-l,4-isoxa- zinc ("morpholine"; Czerminski et al., 1982); and 2-aminopyridine (Bates and Erickson, 1986).
These original papers interpret the reported e.m.f, values in terms of a free hydrogen ion concentration, m(H+), rather than m*(H+) - the problems with this approach are detailed in Dickson (1990a). I have recently used their reported e.m.f, values together with my pub- lished values for E~, to estimate values of Km(BH +) for the conjugate acids of these bases and to assign pm*(H+)s values to equi- molal buffer solutions composed of these various nitrogen bases together with their conjugate acids (Dickson, 1993).
Such buffers can therefore be used to calibrate cell (B) for potentiometric measurements of oceanic pH that will be consistent with the best available values for the acidity constants in sea water (e.g. Khoo et al., 1977b; Dickson, 1990b; Roy et al., 1993-this issue). The sensitivity of such measurements is potentially of the order of about +0.003 pH units (see Fuhrmann and Zirino, 1988), however as a result of the ubiquitous experimental problems alluded to in
the introduction, such measurements will rarely realize an accuracy of better than 0.01 pH units.
5. Basis of spectrophotometric pH measurements
An alternate approach to the measurement of pH in sea water involves the use of indicator dyes together with a high-quality spectrophotometer. For the sulfonephthalein indicators such as thymol blue, m-cresol purple or cresol red, the reaction of interest at sea water pH is the second dissociation
HI-(aq) ~ H+(aq) + I2-(aq) (23)
the indicator being present at a low level in a sea water sample. The total hydrogen ion concen- tration of the sample can then be determined:
pm*(H +) = pK*(HI-) + log[m(IZ-)/m(HI-)] (24)
The principle of this approach uses the fact that the different forms of the indicator have sub- stantially different absorption spectra (Fig. 4). Thus the information contained in the com- posite spectrum (Fig. 5) can be used to estimate m(X2-)/m(HI-).
At an individual wavelength, A, the measured absorbance in a cell with a path length, l, is given by the Beer-Lambert law as:
AA/l = e~(HI-) .m(HI-)
q- £A(I 2- .)m(I 2-) + B~ + e (25)
where B~ corresponds to the background absor- bance of the sample and e is an error term due to instrumental noise. Provided that values of the extinction coefficients, ~A (HI-) and e~ (I 2-) have been measured as a function of wavelength, absorbance measurements made at two or more wavelengths can be used to estimate the ratio m(IZ-)/m(HI-).
In the case that only two wavelengths are used (see e.g. Byrne and Breland, 1989; King and Kester, 1989; Clayton and Byrne, 1993), and provided that the background can be eliminated by a subtractive procedure, Eq. (25) can be rear-
Fig. 4. Extinction coefficients for the three forms of thymol blue indicator (the acid form H2I is present in negligible amounts at sea water pH).
ranged to give (assuming no instrumental error):
m(I 2-)
m(HI-)
A]/ A2 - ~1 (HI-) /e2(HI-)
e] (I2-)/e2(HI-) - (A1/A2)e2(I2-) /e2(HI-)
(26)
the numbers 1 and 2 refer to the two wavelengths chosen. The optimal choice of wavelengths corresponds to using the absorbance maxima of the acid and base forms respectively; this choice is at once the most sensitive as well as being forgiving of minor deviations in wave- length reproducibility.
Thus, calibration of spectrophotometric pH measurements requires the determination of the extinction coefficients of the acid and base forms of the indicator as a function of wavelength - - or at least the extinction coefficient ratios used in Eq. (26) - - as well as the equilibrium constant K*(HI-) as a function of temperature and salinity. Although the extinction coefficients can be measured independently, measurement of K*(HI-) must be done in such a fashion
that the values of m*(H +) obtained from Eq. (24) are compatible with the approach used to define standard reference buffers (see above).
The simplest way to achieve this is to add a small amount of indicator to a standard pH buffer, i.e. one for which values of pm*(H +) have already been assigned using the procedure detailed above. Eqs. (24) and (26) can then be used to estimate values of pK*(HI-) for the indicator. This was the calibration approach used by Clayton and Byrne (1993) who prepared and used the "tris" buffer described by Dickson (1993).
As was mentioned in the previous section, the accuracy of the values of m*(H+)s assigned to such buffers depends on the magnitude of the error that results from assuming that the activity coefficient ratio "y+(HC1)/3,~(HC1) is equal to unity. Clearly this approximation is more likely to hold true at low concentrations of buffer substances in synthetic sea water, i.e. the closer that the buffer composition used approaches that of the pure ionic medium. A series of measurements is now being made in my laboratory in which we prepare a buffer
A.G. Dickson/Marine Chemistry 44 (1993) 131-142
1.0 , ,
139
0.8
0.6
Q
< 0.4
0.2
/
J
0 , 0 i I h
300 400
/ -
• ... / .
I , I
500 600 Wavelength / nm
Fig. 5. Spectrum for thymol blue in a solution of "tris" buffer in
based on synthetic sea water, measure its "total" hydrogen ion concentration using cell (D), then add indicator dye and measure the absorption spectrum. This approach allows us to use low levels of buffer compounds (< 0.01 mol/kg- H20 ) thus enhancing the accuracy of our pH determination, and to use buffers whose pH is optimized to obtain a precise value for K* (HI-).
Consequently, the potential accuracy of spectrophotometric pH measurements is about +0.002, the principal source of error being the original uncertainty in determining Em (Fig. 3). The sensitivity and even the long-term repro- ducibility of spectrophotometric measurements is better than +0.001 pH units. One limitation which needs to be borne in mind is that when one adds an indicator dye to a sea water sample it is possible to change the pH of the sample. Unless care is taken to minimize this effect (see e.g. Clayton and Byrne, 1993), and to take it into account when analyzing the results, this will degrade the experimental accuracy inherent in the spectrophotometric approach.
6. Discussion
A key question in any consideration of pH
700 800
synthetic sea water at 25°C (pH .~ 8.1).
measurements is: how well do we need to know the pH? The answer, of course, is that it depends on the application. We cannot use a pH measure- ment by itself for quantitative information as it is impossible to interpret of itself. To use it as a thermodynamic measurement requires that we know pH - pK*(HB) for the acid-base system of interest - - see Eq. (4). The accuracy in every such application is thus limited not only by how well we can measure pH, but also by how well we know the appropriate equilibrium constants.
In the case of the carbonic acid system in sea water our current knowledge of the equilibrium constants is quite poor in comparison with how well we can measure pH (Dickson and Millero, 1987; Goyet and Poisson, 1989). Nevertheless, new values have recently been reported for these constants (Roy et al., 1993-this issue) and this situation is improving.
The sensitivity of pH measurements (particularly those made using indicator dyes and a high quality spectrometer) is however excellent. They are comparable in sensitivity to our other measurements of sea water carbon dioxide properties, i.e. equivalent to + l#mol/ kg-soln in total dissolved inorganic carbon or
+1 × 10 -6 atm in/(CO2), the fugacity of carbon dioxide in equilibrium with a sample of sea water. Changes in the pH of sea water can thus be interpreted usefully when used together with other information.
We thus have two options for the measure- ment of the total hydrogen ion concentration in sea water: a potentiometric procedure which is based on using cell (B) - - with a glass electrode - - in conjunction with an equation similar to Eq. (12), or a spectrophotometric approach based on Eq. (24).
The potentiometric approach suffers from the electrical problems inherent in making an e.m.f. measurement of a cell with a high internal resistance: drift, susceptibility to noise, pro- blems with ground loops, etc. As a result of such problems, it is desirable to calibrate such measurements frequently using buffers based on synthetic sea water to ensure accuracy. It is however fairly inexpensive to implement this method, requiring only a voltmeter with a high input impedance - - a pH meter - - and a glass electrode. The principal advantage of potentio- metric measurements is that they can provide meaningful data in a non-destructive fashion and at a high reading rate (> 1 Hz).
The spectrophotometric approach, on the other hand, offers freedom from day to day calibration. Its "calibration" is inherent in the prior knowledge of the thermodynamic properties of the indicator dye; these are mea- sured separately in a high-quality laboratory environment. The sensitivity of this approach when used in conjunction with a high-quality double beam spectrophotometer is outstanding: +0.0005 pH units (Clayton and Byrne, 1993) and the accuracy is potentially of the order of +0.003 pH units. Current double beam spectrometers are, however, bulky and expensive, and the sam- ple handling procedures used by Clayton and Byrne to achieve such a high precision result in a sample processing rate of only 8 samples/h.
We need to develop automated sea-going systems which can provide reliable, precise pH values that are consistent with the best available
equilibrium constants for acid-base processes in sea water. Such instruments must be rugged and easy to use. I believe that this can be achieved using current technology, and we are presently in the process of implementing a system which combines a diode array spectrophotometer together with automated mixing of sample and indicator which will be able to measure pH on a flowing stream of sea water to a precision of better than -4-0.001 pH units (Dickson and Pomeroy, 1991).
Such an instrument will, I trust, provide an additional tool in the study of the dynamic bio- geochemical processes which occur in the upper ocean and which may have implications for the control of the climate of the earth. In the future I believe that we will measure pH once more as a routine oceanographic parameter both on surface water and in the deep ocean. The measurement will be easy to make, accurate, and reliable.
A c k n o w l e d g e m e n t s
I should like to acknowledge the many con- tributions of Professor Ric Pytkowicz to this field. His work in the thermodynamics of the carbonate system in sea water contributed to my early interest in this area. I should also like to thank Tonya Clayton for the data presented in Fig. 2 and for access to a prepublication copy of her manuscript.
This work was supported by the National Science Foundation; OCE-9019559.
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