Influence of Local Structural Correlations on Free-Radical Reactions

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Ind. Eng. Chem. Res. 1996, 35, 19-27

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Influence of Local Structural Correlations on Free-Radical Reactions in Supercritical Fluids: A Hierarchical Approach Shankar Ganapathy,† Claude Carlier,‡ Theodore W. Randolph,‡ and James A. O’Brien*,† Department of Chemical Engineering, Yale University, New Haven, Connecticut 06520-8286, and Department of Chemical Engineering, University of Colorado, Boulder, Colorado 80309

We present a general methodology that can be used to study the structural effects of supercritical solvents on fast reactions. Using a hierarchy of theoretical, computational, and spectroscopic methods, we investigate the effect of local solvent structure on a model free-radical reaction. Our results from theory and experiments indicate the existence of locally high solvent densities around reactants at subcritical bulk densities. By comparing simulations and experiments, we show how these local structural correlations enhance the rate constant for Heisenberg spin exchange in supercritical fluids. Introduction Supercritical solvents are appealing candidates for reaction media. The useful solvent properties of supercritical fluids (SCFs) are now well-known and have been characterized in diverse studies [Eckert et al., 1983; Squires et al., 1983; Eckert et al., 1986; McHugh and Krukonis, 1986; Tiltscher and Hofmann, 1987; Russell and Beckman, 1991; Poliakoff and Howdle, 1994]. Current applications of supercritical solvents for reactions include free-radical polymerization [DeSimone et al., 1992], free-radical halogenations of alkyl aromatics [Tanko and Blackert, 1994], isomerization of paraffinic hydrocarbons [Tiltscher et al., 1984; Saim and Subramaniam, 1990; Amelse and Kutz, 1992], and enzymecatalyzed reactions [Randolph et al., 1985; Hammond et al., 1985; Kamihira et al., 1987; Randolph et al., 1988]. Exploring the nature of molecular interactions in SCF mixtures is a necessary first step toward tailoring SCF solvents for reaction processes. Numerous spectroscopic studies of the molecular structure of dilute SCF mixtures [Kim and Johnston, 1987; Kajimoto et al., 1988; Randolph et al., 1988; Yonker and Smith, 1988; Johnston et al., 1989; Brennecke and Eckert, 1988, 1989; Brennecke et al., 1990; Betts et al., 1992; Carlier and Randolph, 1993] have revealed local concentrations of solvent molecules around solutes and of solute molecules around other solutes that differ considerably from bulk values. Efforts to relate microstructure to reactivity have not always borne fruit; some researchers have seen local density effects reflected in reaction rates [Johnston and Haynes, 1987; Zagrobelny and Bright, 1992], while others have not [Roberts et al., 1993]. A comprehensive review of reaction studies under supercritical conditions may be found in Savage et al. (1995). In previous studies, we have attempted to explain why the effect of local structure on reactivity may not be obvious. For instance, several researchers have given theoretical and computational evidence for the existence of strong solute-solute correlations in SCFs [Chialvo and Debenedetti, 1992; Debenedetti and Chialvo, 1992]. Considering these strong interactions, one might expect the local structure to exert a profound effect on reaction rates. In reality, this is not often the case, especially for diffusion-controlled reactions. In fact, we have * To whom correspondence should be addressed. † Yale University. ‡ University of Colorado.

0888-5885/96/2635-0019$12.00/0

shown that, in the limit of diffusion control, the highdensity radial distribution function under reactive conditions shows essentially no short-range structure and, hence, the reaction rate is unaffected by local structure [Randolph et al., 1994, Ganapathy et al., 1995a]. It is well-understood that solute-solute interaction information is reflected in equilibrium solute-solute radial distribution functions (RDFs). While these RDFs suggest vastly different local densities from the bulk values, they do so under nonreactive conditions. We have shown that, in a transport-limited chemical reaction, the time scale for formation of a nonreactive equilibrium solute-solute RDF is always longer than the time scale for reaction [Randolph et al., 1994]. Consequently, the reaction rate is unaffected by the local structure. In this paper, we present a general methodology that can be used to study the effect of the local structure on fast reactions in SCFs. Using a hierarchy of theoretical, computational, and experimental methods, we separate individual molecular interactions, both solvent-solute and solute-solute, and show how these local structural effects in tandem with bulk transport effects can influence reaction rates. We also focus on the phenomenon of solvent-solute clustering that occurs predominantly in dilute SCF mixtures at densities well below the critical density [O’Brien et al., 1993]. These long-lived solvent-solute clusters have been observed in several studies [Carlier and Randolph, 1993; Corti and Debenedetti, 1993; O’Brien et al., 1993] and do not correlate with critical density fluctuations. We illustrate the effect of these solvent-solute correlations on reactions by providing comparisons between electron paramagnetic resonance (EPR) spectroscopic measurements and molecular simulation. Background In order to develop a molecular-level model to predict reactivities, it is rational to combine the strengths of experimental spectroscopy and computational methods. To this end, we have investigated model free-radical reactions using various theoretical and experimental techniques. These techniques are illustrated as follows in order of increasing complexity and physical fidelity: (a) mean-field approaches (such as the kinetic theory of gases and the Stokes-Einstein-Smoluchowski formalism); (b) equilibrium Brownian dynamics (BD) simulations; (c) molecular dynamics (MD) simulations; (d) EPR spectroscopy. © 1996 American Chemical Society

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We use mean-field theories as our base-case description of reaction behavior. At low densities, when the motion of the reacting molecules is ballistic in nature, the kinetic theory of gases provides an accurate description of the rate constant for bimolecular collisions [Hirschfelder et al., 1954]. We note that the reaction rate constant is proportional to the collision rate constant at low densities [Randolph et al., 1994; Ganapathy et al., 1995a]. A modification to the kinetic theory result, accounting for local structure [Randolph et al., 1994; Ganapathy et al., 1995b], gives the collision rate constant the functional form:

( )(

kcol|i,j ) πRc2

8kT π

0.5

)

mi + mj mimj

0.5

gij(r)|r)Rc

(1)

where i and j are the colliding species, Rc is the collision radius, and gij(r)|r)Rc is the equilibrium radial distribution function (RDF) evaluated at the surface of the collision sphere. At high densities, the Smoluchowski relation [Smoluchowski, 1917] relates the reaction rate constant for a diffusional transport-limited reaction to the diffusivity D12 and the collision radius of the reactants:

krxn ) 8πRcD12

(2)

The rate constant can also be expressed in terms of the viscosity:

( )

8kT Rc krxn ) 3η σ

are required, which is on the upper boundary of currently feasible MD simulations. At the highest level of complexity come physical experiments which provide an accurate description of real physical phenomena. We use EPR spectroscopy, which is a sensitive tool for probing local environments in SCF mixtures. The model free-radical reaction we have chosen to investigate is Heisenberg spin exchange between infinitely dilute di-tert-butyl nitroxide (DTBN) radicals. Heisenberg spin exchange is a rapid bimolecular reaction in which two paramagnetic reactants possessing opposite spin states exchange their spins upon collision [Molin et al., 1980]. This is an attractive reaction to study due to its simplicity; reactants and products are chemically identical, differing only in their spin states. Hence, the usual complications associated with time-varying reactant/product concentrations are absent. EPR spectroscopy is particularly useful for studying free-radical reactions such as spin exchange, since it yields both kinetic information (in the form of spin-exchange frequencies) and microstructural information (in the form of nitrogen hyperfine splitting constants). The bimolecular reaction rate constant for spin exchange ke can be represented as the product of the bimolecular collision rate constant kd and the reaction probability p:

kc ) pkd (3)

where σ represents the hydrodynamic diameter of the (identical) reactants. The collision radius (i.e., the distance to which two reactants must approach before reaction can occur) may be quite different from the hydrodynamic radius. The collision radius depends on the reaction chemistry and in most cases is difficult to measure quantitatively. Determination of the hydrodynamic radius, especially in SCFs, is similarly difficult. These low-density and high-density rate constants, when scaled by a suitable reaction probability, provide a starting point for describing the reaction rate over wide density ranges. At the next level of sophistication comes the equilibrium BD technique. BD falls in the category of stochastic dynamics techniques wherein some degrees of freedom are constrained by stochastic bounds [Chandrasekhar, 1943]. Solute-solute interactions are explicitly dealt with using this technique. The solvent, however, is represented by a mean field, and the influence of solvent-solute interactions is limited to Brownian noise. Consequently, all effects of solvent structure on reactivity are neglected. In the current work, this limitation has been turned to our advantage since any differences between experiments and BD simulations are, at least in part, attributable to the influence of solvent-solute interactions on reactivity. MD represents the “high end” of simulation techniques, where we get the exact solution to an ideal problem, constrained only by the degree of realism of the intermolecular potential used. Both solvent-solute and solute-solute correlations and their effect on reaction rates can be studied using this technique. However, the high computational costs of MD curtail its usefulness. Simulating reactions at conditions of nearinfinite dilution (say, a mole fraction of 0.001) requires a thousand solvent molecules for each solute. To obtain reasonable statistics for solute-solute phenomena, then, systems as large as a few hundred thousand molecules

(4)

The reaction probability p is related to the average lifetime τc of each collision, the exchange integral J which is a molecular property of the free radical, and the spin states of the reactants. For the case of DTBN which is spherical and possesses a spin state of 1/2, the probability is given by [Currin, 1962]:

p)

[

2 2 1 J τc 2 1 + J2τ 2 c

]

(5)

We use the aforementioned hierarchy of techniques to study spin exchange between DTBN probes in three different solvents; two of them are nonpolar (carbon dioxide and ethane), and one is moderately polar (trifluoromethane). Note that trifluoromethane (CHF3) or fluoroform possesses a permanent dipole moment of 1.65 D [Kajimoto et al., 1988], while carbon dioxide (CO2) has a quadrupole moment of 4.1 × 108 D‚cm [Prausnitz, 1969], and ethane (C2H6) has no dipole moment. Materials and Methods EPR Spectroscopy. A high-pressure closed-loop reactor system was used for all of the experiments, details of which may be found elsewhere [Randolph and Carlier, 1992]. Multiparameter equations of state were used to calculate the densities of CHF3 (Air Products, 99.95%) [Aizpiri et al., 1991], CO2 (Air Products, 99.99%) [Angus et al., 1976], and C2H6 (Matheson, 99.0%) [Younglove and Ely, 1987]. Each experiment was carried out at a range of constant concentrations of DTBN (Sigma, 99.0%) and at two different temperatures corresponding to Tr ) 1.01 and 1.08. A pressure range from Pr ) 0.6 to 1.6 was investigated for the three solvents. A DTBN concentration range from 0.4 to 2.0 mM was explored in different experiments; the concentration in any one experiment, however, remained constant. Measurements were carried out in PEEK tubing (Alltech). PEEK has a characteristic EPR signal which must be subtracted from the recorded spectra.

Ind. Eng. Chem. Res., Vol. 35, No. 1, 1996 21

Figure 1. Radial distribution functions for solvent around solvent (g11), solvent around solute (g12), and solute around solute (g22) calculated from the OZPY equation at a reduced density Fr ) 0.5 and reduced temperature Tr ) 1.01. The L-J parameters of DTBN and ethane were used in this calculation. r* is in solvent L-J units. Figure 3. Reaction rate constant ke (L mol-1 s-1) as a function of reciprocal reduced density 1/Fr for spin exchange for DTBN in the following. (a, b) CHF3 at DTBN concentrations ranging from 0.6 to 1.2 mM: (a) squares, 0.6 mM; inverted triangles, 1.2 mM; (b) circles, 0.9 mM; inverted triangles, 1.2 mM. (c, d) CO2 at DTBN concentrations ranging from 0.6 to 0.9 mM: (c) squares, 0.6 mM; inverted triangles, 0.8 mM; circles, 0.9 mM; (d) inverted triangles, 0.6 mM; circles, 0.9mM. (e, f) Ethane at DTBN concentrations of 0.1 and 0.5 mM: (e) circles, 0.1 mM; (f) circles, 0.5 mM. Each pair of graphs represents spin exchange in the same solvent at two different temperatures of Tr ) 1.01 and 1.08.

Figure 2. Surface of local density enhancement of solvent around solute as a function of the radial separation r* (normalized in solvent diameters), and the bulk density F [M] for the system DTBN-ethane. This calculation was done at a reduced temperature Tr ) 1.01. Table 1. Mixed Component L-J Parameters for DTBN DTBN-carbon dioxide DTBN-ethane DTBN-fluoroform

σ12 (Å)

12 (kcal/mol)

6.4476 6.4736 6.89

0.4537 0.5596 0.578

X-band EPR spectra were recorded in a Bruker ESP 300E spectrometer. Spectra were recorded using a microwave power of 1.0 mW, a modulation amplitude of 1.01 G, a time constant of 0.64 ms, and a total scan time of 20.97 s. The center field was fixed at 3480 G, and a scan width of 75 G was used. Measured spectra were fitted using programs [Carlier, 1994] based on the line-shape simulation programs of Schneider and Freed (1989). The fitted parameters were the spin-exchange frequency and the nitrogen hyperfine splitting constant AN. Corrections for spin-rotation interactions [Atkins and Kivelson, 1966] were made based on line-width measurements at very low DTBN concentrations (10-5 M). Molecular Simulation. Equilibrium BD reaction simulations were carried out over a wide range of densities. More information about the BD technique as adapted to studying reactions may be found in Allen (1980) and Ganapathy et al. (1995a). The LennardJones (L-J) intermolecular potential was used to model the reactants. L-J parameters for ethane [Prausnitz,

Figure 4. Nitrogen hyperfine splitting constant AN [G] vs reduced density Fr for DTBN in CHF3 (a, b), CO2 (c, d), and ethane (e, f) at two different temperatures (Tr ) 1.01 and 1.08). Nonlinearities in AN indicate that local densities around the nitroxide probe are considerably different from the bulk values.

1969], fluoroform [Aizpiri et al., 1991], and carbon dioxide [Hirschfelder et al., 1954] were taken from the literature. The L-J parameters for DTBN in each of the three solvents were determined using group contribu-

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a

b

c

Figure 5. Local density enhancement as a function of reduced density Fr for (a) CO2 around DTBN, (b) ethane around DTBN, and (c) CHF3 around DTBN estimated from EPR measurements at Tr ) 1.01 (closed circles) and 1.08 (closed diamonds), integral equation calculations at Tr ) 1.01 (open triangles), and MD simulations at Tr ) 1.01 (open squares). A cutoff of rmax ) 1 solute + 1 solvent diameter has been used to calculate local densities from MD and integral equation RDFs.

tion methods [Jorgensen et al., 1984; Jorgensen and Tirado-Rives, 1988; Jorgensen et al., 1990] and a spherical averaging technique [O’Brien et al., 1993]. These parameters are listed in Table 1. Pure-component L-J parameters for DTBN in all three solvents were calculated using geometric mixing rules consistent with the approach used by Jorgensen and co-workers [1984, 1988, 1990]. Solute diffusivities, which are an input to BD, were estimated using both NVE and NVT MD simulations of lengths of up to 4.5 × 105 time steps (dimensionless ∆t ) 0.003) [Ganapathy et al., 1995a]. BD simulations were run for as long as 3 × 106 time steps (dimensionless ∆t ) 0.0064), with

the first 20 000 steps allowed for equilibration. The MD diffusion coefficients were estimated using a single solute molecule in 1000 solvent molecules; since no solute-solute interactions were therefore possible, this was effectively infinite dilution. BD runs were carried out over a solute mole fraction range from 0.001 to 0.01. Results and Discussion Figure 1 shows the solvent-solvent (g11), solventsolute (g12), and solute-solute (g22) RDFs calculated from the Percus-Yevick closure to the OrnsteinZernike (OZPY) equation [McGuigan and Monson,

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1990]. This calculation is for a near-infinite dilution L-J fluid mixture at Tr ) 1.01 and Fr ) 0.7, using the L-J parameters for DTBN and ethane. Figure 2 shows a surface of the local density enhancements of solvent around solute as a function of bulk density and radial separation. These local densities were calculated from the appropriate modified fluctuation integral: local F12 (r) bulk F12

∫rr

3 )1+

r2[g12(r) - 1] dr

max

min

(r3 - rmin3)

(6)

where rmin corresponds to the spatial separation where the RDF becomes significantly nonzero. As can be seen, the maximum in the local density of solvent around solute occurs at a bulk density close to 1/2 the critical density (6.875 M for ethane). Such augmentations in the local density are a short-ranged effect [Knutson et al., 1992; O’Brien et al., 1993; Carlier and Randolph, 1993] and do not scale with the isothermal compressibility. Parts a-f of Figure 3 illustrate the reaction rate constant for spin exchange ke as a function of reciprocal reduced density in fluoroform, carbon dioxide, and ethane, respectively. Each pair of graphs represents measurements made at two different temperatures corresponding to Tr ) 1.01 and 1.08. Above the critical density, the reaction rates decrease sharply with increasing density, as is to be expected for a reaction approaching diffusion control. The nitrogen hyperfine splitting constant AN is an accurate probe of local solution structure in the cybotactic region [Carlier and Randolph, 1993]. The unpaired electron density on the nitroxyl nitrogen increases with solvent polarity, resulting in larger values of AN. This makes AN a useful parameter for measuring local solvent densities. Parts a-f of Figure 4 show AN vs reduced density in fluoroform, carbon dioxide, and ethane, respectively. Plots of hyperfine splittings vs density show considerable deviations from linearity in all of the solvents. The first step toward calculating local densities from AN values is to relate hyperfine splittings to the dielectric constant using the solvatochromic theory of McRae [McRae, 1956]. Dielectric constants may be related to the density through the Clausius-Mossotti relation [Carlier and Randolph, 1993]. There is no direct correlation between AN and the dielectric constant available in the literature for hydrogen-bonding dipolar solvents such as fluoroform. We calculated local densities for fluoroform using the relation between AN and the Kamlet-Taft π* parameter [Kamlet and Taft, 1979]. Consequently, the uncertainty associated with the local density enhancement (LDE) is greater for fluoroform than for carbon dioxide and ethane whose solvatochromic parameters are well tabulated [Johnston and Cole, 1962; Younglove and Ely, 1987; Carlier and Randolph, 1993]. The ratio of the local density calculated from AN values to the bulk density of CO2 is shown in Figure 5a as a function of the bulk density (in reduced units) for two different temperatures. Also shown in Figure 5a are the local density enhancements (LDEs) calculated from solvent-solute RDFs obtained from (a) MD simulations and (b) the OZPY equation. L-J parameters that mimic the DTBN-CO2 system have been used in the simulations and integral equation calculations. A cutoff value of rmax ) 1 solute + 1 solvent diameter has been

Figure 6. Average collision lifetimes (ps) from BD simulations as a function of bulk density F (M) for DTBN in ethane.

Figure 7. Reaction rate constant ke (L mol-1 s-1) as a function of reciprocal reduced density 1/Fr for spin exchange in CHF3 . The closed symbols represent EPR measurements made at different concentrations, while the open circles denote BD simulations. Table 2. Values of the Exchange Integral J for DTBN J (s-1) DTBN in carbon dioxide DTBN in ethane DTBN in fluoroform literature value [Molin et al., 1980]

1.22905 × 1011 5.63354 × 1010 9.28403 × 1010 6.0 × 1010

used for the LDE calculations based on RDFs from MD and the OZPY equation. It can be seen that LDEs from both EPR and theoretical predictions reach a maximum that occurs at a density that is close to 1/2 the critical density. Similar trends are observed in the local densities of DTBN around ethane and fluoroform (see Figure 5b,c). Note that the magnitude of the local density enhancement in fluoroform is unlikely as it predicts a value that is above liquid density. This may be attributed to the approximations we have made in that specific calculation, as mentioned above. The question to be asked then is, do these observed

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a

b

c

Figure 8. Ratio of reaction rate constants from EPR to fitted reaction rate constants from BD simulations (closed triangles) vs reduced density Fr for the following: (a) Spin exchange in ethane. The open diamonds denote the ratio of EPR rate constants to reaction rate constants from BD runs that incorporated the reaction probability directly in the simulations. There is good agreement between the two sets of normalized rate constants. (b) Spin exchange in CO2. (c) Spin exchange in CHF3. Also plotted are the corresponding local density enhancements (closed circles) from EPR measurements. For all three solvents, the maximum in the normalized reaction rate constant occurs at the same bulk density as the maximum in the local density.

solvent-solute LDEs affect reaction rates? We attempt to answer this question by comparing reaction rate constants from EPR with “reaction” rate constants from BD simulations. The isotopic reaction scheme used in the BD calculations [Randolph et al., 1994] presupposes a reaction probability of unity; this ideal reaction rate has to be scaled by an appropriate reaction probability to allow quantitative comparison with experimental rate constants. Alternatively, if the reaction probability for

spin exchange is known a priori, one can incorporate this probability directly in the BD simulation and calculate an effective reaction rate constant. The probability for spin exchange is given by eq 5 and is a function of both the average collision lifetime τc and the exchange integral J. As a first step toward determining the probability, we calculated average collision lifetimes from BD as a function of density (see Figure 6). Since the definition of a collision lifetime depends on the size

Ind. Eng. Chem. Res., Vol. 35, No. 1, 1996 25

of the collision sphere, we must specify the reaction radius Rc. For most reactions, the reaction radius does not correspond to a precise numerical value; rather, it is a weighted average of a distribution of collision radii. The rate of spin exchange is known to exhibit a r-6 radial dependence [Molin et al., 1980]. At a collision radius of Rc ) 1 solute + 1 solvent diameter, more than 95% of spin exchange occurs inside the collision sphere. As a first approximation, we have used this collision radius for all of our calculations. Note that the collision lifetimes display only a weak dependence on solvent density except at very low densities. Consequently, reaction probabilities calculated from these lifetimes do not vary widely. For the DTBN-ethane system, we computed rate constants from BD simulations using both of the routes discussed above: reaction probability imposed either (i) by scaling the results of a unity reaction probability simulation after the simulation or (ii) by applying it directly at each collision event during the simulation. In the former (a posteriori) method, we fitted the reaction probability using the exchange integral J as a fitting parameter, by forcing the experimental and (unity reaction probability) simulated reaction rates to match for the four highest density BD points, using collision lifetimes τc measured during the simulation (See eqs 4 and 5). The resulting fitted exchange integral values are in good agreement with literature values [Molin et al., 1980] for spin exchange in all three solvents as shown in Table 2. In the latter (a priori) technique, we first computed a reaction probability using the exchange integral value fitted in the previous step and used it as a direct input to the BD simulations. In the simulations, a random number routine that produced uniform deviates in (0, 1) was used to generate a number every time a collision between two reactants occurred. The success or failure of the event (i.e., reaction) was decided by comparing this number with the specified probability [Ganapathy et al., 1995c]. Reaction rate constants calculated using the a priori and the a posteriori methods were identical. Figure 7 illustrates the reaction rate constants from EPR measurements along with the scaled rate constants from BD simulations as a function of the reciprocal reduced density for spin exchange in fluoroform. Reaction rates from BD agree quantitatively with EPR measurements at high densities. At lower densities, however, the BD rates are considerably different from the EPR reaction rates. The disagreement appears to be largest at densities near 2/3 the critical density. To highlight more clearly the differences between experimental and computational rates, we plotted the EPR reaction rates, normalized by the corresponding BD predictions as a function of density, for spin exchange in CO2 (Figure 8a). Also plotted in Figure 8a is the corresponding local solvent-solute density enhancement as a function of density, calculated from AN measurements. Again, while the EPR and BD rates agree well at the high densities, they differ considerably at lower densities. Since BD simulations incorporate solute-solute interactions and neglect solvent-solute interactions, any differences between BD and experiments are likely to be a manifestation of solvent-solute interactions. Remarkably, the maximum in the normalized reaction rate occurs at the same density as the maximum in the local density enhancement. This suggests that local density enhancements of solvents around solute molecules increase the reaction rate

constant for spin exchange. This conclusion is corroborated by spin exchange measurements in the other two solvents; normalized reaction rates for spin exchange in ethane and fluoroform are considerably different from unity at the lower densities, and the maxima in the normalized rate constants correspond to the maxima in the local solvent-solute density augmentations in these solvents (as illustrated in Figure 8b,c). Conclusions We have shown that theory and experiment complement each other when used in a hierarchical fashion, and this may be a useful way to infer local structural effects on SCF reactivity. Using mean-field theories, BD simulations, MD simulations, and EPR spectroscopy, we have probed the effect of local solvent structure on the reaction rate for Heisenberg spin exchange. Measured hyperfine splittings from EPR spectroscopy indicate the existence of locally high solvent densities around solutes at bulk densities that are below the critical density. By providing comparisons between spectroscopy and simulations, we have shown that the locally high solvent densities around solutes enhance the reaction rate for Heisenberg spin exchange. Observed local density enhancements and the corresponding maxima in normalized reaction rates occur at densities well below the critical density. Short-ranged phenomena such as chemical reactions are influenced by short-ranged local density effects and do not scale with long-ranged critical effects. Acknowledgment is made to the National Science Foundation for its support through the Presidential Young Investigator Award and Grant CTS 9414759 to T.W.R. and Grant CBT-8809548 to J.A.O. Nomenclature AN ) nitrogen hyperfine splitting constant (G) BD ) Brownian dynamics CO2 ) carbon dioxide CHF3 ) trifluoromethane, or fluoroform C2H6 ) ethane DTBN ) di-tert-butyl nitroxide EPR ) electron paramagnetic resonance spectroscopy J ) exchange integral (s-1) L-J ) Lennard-Jones MD ) molecular dynamics NVE ) conditions of a constant number of molecules, volume, and internal energy (microcanonical ensemble) NVT ) conditions of a constant number of molecules, volume and temperature (canonical ensemble) D ) diffusion coefficient g(r) ) radial distribution function k ) Boltzmann constant ke ) EPR reaction rate constant (L mol-1 s-1) kcol ) collision rate constant krxn ) reaction rate constant m ) molecular mass p ) reaction probability r ) radial distance, spatial position Rc ) collision radius ∆t ) time step T ) temperature Greek Symbols  ) L-J energy parameter (kcal mol-1) η ) viscosity F ) density

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σ ) L-J length parameter (Å) τ ) collision lifetime (ps) Subscripts and Superscripts 11 ) property of solvent, component 1 22 ) property of solute, component 2 12 ) solvent-solute cross property r ) reduced property * ) dimensionless property/parameter c ) collision-related property

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Received for review May 1, 1995 Revised manuscript received September 7, 1995 Accepted September 19, 1995X IE950272O

X Abstract published in Advance ACS Abstracts, November 15, 1995.