3354
J. Phys. Chem. B 1998, 102, 3354-3362
Molecular Force Measurement in Liquids and Solids Using Vibrational Spectroscopy Erik J. Hutchinson and Dor Ben-Amotz* Purdue UniVersity, Department of Chemistry, West Lafayette, Indiana 47907-1393 ReceiVed: September 18, 1997; In Final Form: NoVember 25, 1997
Shifts in molecular vibrational frequencies are used to measure intermolecular forces in liquids and solids as a function of external pressure. The force along a particular bond within a molecule is derived from its measured vibrational frequency shift using an expression for the perturbation of a quantum anharmonic oscillator in a classical bath. New pressure induced frequency shift and force measurements are performed on the CdC bond in 1-octene, trans-2-octene and trans-4-octene (in both pure liquids and methanol solutions), and on the Si-O bond in three methylsiloxanes. Comparison of these and previous gas-to-liquid (or solid) and high-pressure vibrational frequency shift results reveal a large variation in the force on different types of bonds, while families of similar bonds experience a similar force at a given external pressure, with only a weak dependence on the location of the bond within the solute or the molecular structure of the surrounding solvent. Physical interpretations of the results using both continuum and perturbed hard sphere fluid models are suggested.
I. Introduction An isolated molecule is one that, by definition, exists in an environment free of external forces. In a condensed phase material, on the other hand, the forces between molecules determine both the molecular and the bulk properties for the system. Thus the forces which mediate intermolecular interactions offer a window into the relationship between the macroscopic properties and molecular structure of liquids and solids. Here we report the development and application of a molecular force gauge, used to measure forces exerted on individual bonds within a molecule. The capabilities of this technique are demonstrated using the analysis of solvation and pressure induced intermolecular forces in liquids and solids. This new spectroscopic force measurement technique compliments recent advances in atomic force microscopy (AFM), which measures forces on an atomically sharp tip near a surface. AFM has been widely used to map surface morphology, friction, and wear on a molecular scale, as well as to manipulate atoms and molecules.1 The forces exerted and measured using AFM are typically in the nano-Newton (nN) range. Our spectroscopic force measurement technique differs from AFM in having higher sensitivity, as forces as small as several pico-Newtons (pN) may be measured. Furthermore, unlike AFM, this technique is noninvasive as opposed to manipulative, in that it measures the intrinsic forces between molecules rather than forces produced by the measurement apparatus. The new technique may also be related to force measurements inferred from flow induced degradation (covalent bond cleavage) of polymers.2-4 Such experiments reveal that in a sufficiently strong velocity-gradient long-polymer molecules tend to break, usually into two equivalent shorter chains. An analysis of the strain rates required to break bonds as a function of polymer chain length suggests that about 5-10 nN of force are required in order to break a single covalent bond, which is in good agreement with the maximum intramolecular force derived from the slope of Morse potential used to model the corresponding covalent bonds. Unlike such indirect force measurements, the present force gauge can measure forces of the order of 1000
times smaller in magnitude than those required to break a bond. Furthermore, this technique can measure forces that are either positive (compressional) or negative (elongational), while bond cleavage clearly results from an elongational force. Our spectroscopic force gauge is based on the relationship between the force experienced along a molecular bond and the associated change in the frequency of a vibrational mode involving that bond. In other words, the forces between molecules produce changes in molecular vibrational frequencies which may be measured using IR or Raman spectroscopy. Such a coupling between intermolecular interactions and intramolecular vibrations was clearly recognized long ago by David Buckingham when he derived a second-order perturbation expression for the vibrational frequency shift of a quantum anharmonic oscillator, resulting from its interactions with a bath solvent molecules.5-8 The present work in essence derives from an inversion of Buckingham’s frequency shift results, so as to express the magnitude of an intermolecular force along a bond as a function of the corresponding shift in vibrational frequency. The remainder of this paper begins, in section II, with a description of two simple physical pictures for solvation-induced forces, one in a hydrodynamic continuum solvent, and the other in a fluid composed of hard spheres (with or without a cohesive mean field). Section III describes the approximations made in deriving expressions for molecular force as a function of vibrational frequency shift. Section IV describes the experimental method used to measure pressure induced Raman scattering frequency shifts. Section V contains a summary of the results and a discussion of general trends revealed in the forces derived from gas-to-liquid and pressure induced vibrational frequency shifts. General conclusions are collected in section VI. II. Physical Interpretation of Solvation Forces An appealing simple physical picture for the relationship between molecular vibrations and an applied force may be obtained from the fundamental mechanical relation between the force acting along a bond F and the associated potential energy
S1089-5647(97)03065-4 CCC: $15.00 © 1998 American Chemical Society Published on Web 04/15/1998
Molecular Force Measurement
J. Phys. Chem. B, Vol. 102, No. 18, 1998 3355
U, external pressure P, and volume V, with a sign convention chosen so that the force is positive when expanding against a positive external pressure.
F ) -dU/dr ) P dV/dr
(1)
Thus the pressure derivative of a force along a diatomic bond is simply equal to the derivative of the volume with respect to distance (which in this case is the bond length of the diatomic).
dF/dP ) dV/dr
(2)
Notice that this pressure derivative has units of area (volume/ length). Thus a pressure derivative of 100 pN/GPa (corresponding to a typical pressure derivative of the force on a CdC bond) represents an area of 10 Å2, which is remarkably close to the van der Waals cross-sectional area of a carbon atom, π(1.7 Å)2 ) 9 Å2. A somewhat more sophisticated way to model the above pressure derivative can be obtained using an idealized solution composed of a hard diatomic dissolved in a hard sphere fluid.9 For example, taking the diameters and bond length of the diatomic to be those appropriate for the 1-octene CdC bond (σ1) σ2 ) 3.4 Å and re ) 1.31 Å) and a solvent hard sphere diameter appropriate for methanol (σS ) 3.84 Å), at the density and temperature of liquid methanol, yields a pressure-induced force of 200 pN/GPa, which compares reasonably well with the experimental pressure derivative of 120 pN/GPa (particularly considering the fact that attractive interactions, which have been entirely neglected in the hard body fluid model, would tend to decrease the predicted pressure derivative). More generally, in a system which includes both repulsive and attractive (cohesive) intermolecular interactions, the pressure derivative of the force along a solute bond represents a measure of the change in volume of the system (solute plus solvent) along the solutes vibrational coordinate.10,11 This volume change is driven by the change in the solute’s solvation energy upon vibration excitation. When attractive solvation forces dominate (as is typically true in the vapor or low density fluid phase), the solvent tends to contract about the vibrationally excited solute, producing a decrease in the system volume and a negative (elongational) force along the solute bond.5 When intermolecular repulsive interactions dominate (as is often the case in high-pressure liquids) the converse is true, and a positive (compressive) force is produced along the solute bond.9 III. Force Measurement Theory In deriving the effect of intermolecular interactions on molecular vibrations, Buckingham began by expanding the interaction potential energy Uint between a diatomic solute and a solvent, as a function of the displacement of the solute from its equilibrium bond length re.6-8
Uint ) U0 + U′(r - re) + 1/2U′′(r - re)2
(3)
Such a force field was shown to produce the following shift in the vibrational frequency of the solute, where νe is the classical harmonic frequency and f and g are the harmonic and anharmonic force constants of the isolated solute, respectively.5,9
∆ν )
[( )
]
g νe -3 〈U′〉 + 〈U′′〉 2f f
(4)
The angle brackets represent a Boltzmann average over the equilibrium configurations of the solvent bath. Thus 〈U′〉
TABLE 1: Experimental Pressure P, CdC Frequency Shift, ν (Relative to Liquid at 1 atm), and Density G (at 1 atm), for Octenes 1-octene F ) 0.715 g/mL trans-2-octene F ) 0.718 g/mL trans-4-octene F ) 0.715 g/mL 10% wt 1-octene/MeOH
P (GPa) ν (cm-1) P (GPa) ν (cm-1) P (GPa) ν (cm-1) P (GPa) ν (cm-1) P (GPa) ν (cm-1) P (GPa) ν (cm-1)
10% wt trans-2-octene/MeOH 10% wt trans-4-octene/MeOH
0.31 0.93 0.03 0.02 0.14 0.47 0.18 1.06 0.18 0.80 0.13 0.13
0.61 2.57 0.17 0.82 0.40 1.60 0.59 2.54 0.60 2.23 0.20 0.61
1.02 3.72 0.53 2.03 0.71 2.73 1.15 5.03 0.92 3.47 0.75 2.78
1.61 6.55 0.98 4.12 1.01 4.27 2.04 8.78 1.36 5.42 1.24 4.97
represents the solvent configuration averaged mean force along the solute bond. The second-order force 〈U′′〉 may be expressed as a sum of the derivative of the mean force and a term proportional to the force fluctuation.5 Although the latter is more difficult to treat theoretically, David Oxtoby12 has suggested the following approximate relation between the first and second-order force (following earlier work of Schwartz, Slawsky, and Herzfeld),9,13-15 where σC ) (σ1 + σ2 + 2σS)/2 is the average interaction diameter of the solute atoms (of diameters σ1 and σ2) and the solvent (of effective hard sphere diameter σS).16,17
〈U′′〉Oxtoby ≈
17.5 〈U′〉 t h〈U′′〉 σC
(5)
Although the validity of this approximation, when applied to condensed phase frequency shifts, may be questioned,9 it suggests the following approximate relation between a measured frequency shift ∆ν and the force F along the corresponding bond.
F ) 〈U′〉 ≈
[
2f
]
νe(h - 3g/f )
∆ν
(6)
Note that even if the second-order force is neglected (h ) 0), the predicted proportionality between F and ∆ν still holds, in this case with a coefficient which contains only parameters of the solute.
F≈
[ ] -2f 2 3νe g
∆ν
(7)
Since h is predicted to be positive (and g/f is negative), the magnitude of the forces derived from eq 6 are invariably smaller than those derived from eq 7. Given the uncertainty in the actual size of the second-order force,9 the difference between the forces predicted by eqs 6 and 7 may be taken as a rough measure of the absolute uncertainty in F (which is of the order of 30%, as shown by the F6/F7 values in Table 3 and discussed in section V). On the other hand, either one of the above equations may be expected to reliably predict relatiVe changes in force as a function of pressure, or those in systems with different solvent or solute molecules. Since the above results pertain to the frequency shift of a diatomic solute molecule, some generalization or further approximation is required in applying these results to polyatomic normal-mode frequency shifts. One approach has been suggested by Schweizer and Chandler,18 who treated solventinduced forces on polyatomic vibrations as a superposition of isolated diatomic bond vibrations weighted to reflect their normal mode amplitudes. This approach, although not exact,
3356 J. Phys. Chem. B, Vol. 102, No. 18, 1998
Hutchinson and Ben-Amotz
TABLE 2: Molecular Parameters (ν0, re, σ1, σ2, σS, f, A ) reg/f) sample C-C stretch CH3CN CH3CN CH3CN/CH2Cl2 CH3CN/CHCl3 CH3CN/MeOH ethane/CH2Cl2 acetone CdC stretch 2-methylpropene 1-octene 2-octene 4-octene 1-octene/MeOH 2-octene/MeOH 4-octene/MeOH C ring breathing pyridine/n-octane pyridine/isooctane pyridine/CH2Cl2 pyridine/acetone pyridine/H2O C-H stretch ethane/CH2Cl2 CH3CN CH3CN CH3CN/CCl4 CH4 MeOH MeOH CH3I CH3I C-D stretch CD3I O-H stretch MeOH MeOH/CS2 n-BuOH/CS2 t-BuOH/CS2 N-H stretch NH3 C-O stretch MeOH MeOH superpressed CdO stretch acetaldehyde acetone acetone/cyclohexane propylene carbonate CtN stretch CH3CN CH3CN CH3CN/CCl4 CH3CN/methanol NtN stretch N2 C-I stretch CH3I CD3I CH3I CHI3 I-I stretch I2/MCH I2/n-hexane I2/n-octane I2 F-F stretch F2 OdCdO stretch carbon dioxide SdCdS stretch CS2 Si-O-Si stretch HMDS OMTS DMTS
phase
ν0 (cm-1)
re (Å)
σ1 (Å)
σ2 (Å)
(σS) Å
f (N/cm)
A re(g/f)
s g g g g g l
921 920.2 920.2 920.2 920.2 994.8 780
1.47 1.47 1.47 1.47 1.47 1.54 1.51
3.4 3.4 3.4 3.4 3.4 3.4 3.4
3.4 3.4 3.4 3.4 3.4 3.4 3.4
4.24 4.24 4.64 5.05 3.84 4.64 4.85
5.16 5.26 5.26 5.26 5.26 4.87 4.4
-2.57 -2.57 -2.57 -2.57 -2.57 -2.5 -2.98
29, 31 19 19 19 19 20 30, 31
g l l l l l l
1661.1 1642.2 1673 1669.4 1642.2 1673 1669.4
1.31 1.31 1.31 1.31 1.31 1.31 1.31
3.4 3.4 3.4 3.4 3.4 3.4 3.4
3.4 3.4 3.4 3.4 3.4 3.4 3.4
5.05 6.47 6.46 6.46 3.84 3.84 3.84
15.30 15.47 15.23 15.32 15.47 15.23 15.32
-2.76 -2.14 -2.16 -1.95 -2.14 -2.16 -1.95
31 this work this work this work this work this work this work
g g g g g
991.4 991.4 991.4 991.4 991.4
1.42 1.42 1.42 1.42 1.42
3.4 3.4 3.4 3.4 3.4
3.4 3.4 3.4 3.4 3.4
6.55 6.54 4.64 4.85 2.92
7.67 7.67 7.67 7.67 7.67
-2.64 -2.64 -2.64 -2.64 -2.64
9 9 9 9 32
g g s g s l s s g
2953.8 2954.1 2945 2954.1 2914 2836 2844 2950 2969
1.10 1.11 1.06 1.11 1.12 1.12 1.08 1.09 1.09
3.4 3.4 3.4 3.4 3.4 3.4 3.4 3.4 3.4
2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4
4.64 4.24 4.24 5.39 3.59 3.84 3.84 4.59 4.59
4.74 4.88 5.33 4.88 5.44 4.48 4.8 4.86 4.86
-2.21 -2.26 -2.26 -2.26 -2.43 -2.08 -2.08 -2.19 -2.19
20 19 29 19 33, 31 34, 31 35, 31 36, 31 20
s
2143
1.09
3.4
2.4
4.59
4.86
-2.19
36, 31
g l l l
3681 3630 3624 3607
0.97 0.97 0.97 0.97
3.04 3.04 3.04 3.04
2.4 2.4 2.4 2.4
3.84 4.52 4.52 4.52
7 7.8 7.8 7.8
-2.16 -2.16 -2.16 -2.16
35, 31 37, 31 37 37
l
3217
1.01
3.1
2.4
2.98
5.97
-2.03
38, 31
g g
1033 1033
1.41 1.41
3.4 3.4
3.04 3.04
3.84 3.84
19.02 19.02
-2.83 -2.83
35, 31 35, 31
g g l l
1743 1731 1721 1781.3
1.21 1.21 1.21 1.21
3.4 3.4 3.4 3.4
3.04 3.04 3.04 3.04
4.22 4.85 4.85 5.29
12.9 12.9 12.9 13
-2.71 -2.71 -2.71 -2.71
31 39, 31 39, 31 40
g s g g
2266.5 2254 2266.5 2266.5
1.16 1.14 1.16 1.16
3.4 3.4 3.4 3.4
3.1 3.1 3.1 3.1
4.24 4.24 5.39 3.84
17.73 18.33 17.73 17.73
-2.77 -2.77 -2.77 -2.77
19 29 19 19
g
2330
1.10
3.1
3.1
3.45
22.95
-2.71
28
ref
s s g s
530 500 532.8 574
2.16 2.16 2.16 2.16
3.4 3.4 3.4 3.4
3.96 3.96 3.96 3.96
4.59 4.59 4.59 5.88
2.28 2.28 2.28 2.28
-3.28 -3.28 -3.28 -3.28
36, 31 36, 31 20 42
g g g s
213.3 213.3 213.3 178.6
2.67 2.67 2.67 2.67
3.96 3.96 3.96 3.96
3.96 3.96 3.96 3.96
6.02 5.96 6.55 4.84
1.72 1.72 1.72 1.72
-4.01 -4.01 -4.01 -4.01
9 9 9 43
s
1110
1.42
2.94
2.94
3.03
3.5
-3.68
44
s
1384
1.21
3.4
3.04
3.38
12.9
-2.71
41, 31
s
658
1.55
3.4
3.6
4.52
7.88
-3.00
45, 31
l l l
519.8 507.0 495.8
1.64 1.632 1.63
3.04 3.04 3.04
4.2 4.2 4.2
6.96 7.83 8.55
8.04 8 8
-2.11 -2 -2
this work this work this work
Molecular Force Measurement
J. Phys. Chem. B, Vol. 102, No. 18, 1998 3357
TABLE 3: Gas-to-Liquid and Pressure-Induced Frequency Shift and Force Results sample C-C stretch CH3CN (solid) CH3CN CH3CN/CH2Cl2 CH3CN/CHCl3 CH3CN/MeOH ethane/CH2Cl2 acetone CdC stretch 2-methylpropene 1-octene 2-octene 4-octene 1-octene/MeOH 2-octene/MeOH 4-octene/MeOH C ring breathing pyridine/n-octane pyridine/isooctane pyridine/CH2Cl2 pyridine/acetone pyridine/H2O C-H stretch ethane/CH2Cl2 CH3CN CH3CN (solid) CH3CN/CCl4 CH4 (solid) MeOH MeOH (solid) CH3I (solid) CH3I C-D stretch CD3I (solid) O-H stretch MeOH (solid) MeOH/CS2 n-BuOH/CS2 t-BuOH/CS2 N-H stretch NH3 C-O stretch MeOH (solid) MeOH superpressed CdO stretch acetaldehyde acetone acetone/cyclohexane propylene carbonate CtN stretch CH3CN CH3CN (solid) CH3CN/CCl4 CH3CN/methanol NtN stretch N2 C-I stretch CH3I (solid) CD3I (solid) CH3I CHI3 (solid) I-I stretch I2/MCH I2/n-hexane I2/n-octane I2 (solid) F-F stretch F2 OdCdO stretch carbon dioxide (solid) SdCdS stretch carbon disulfide (solid) Si-O-Si stretch HMDS OMTS DMTS
νg-l (cm-1) 1 1 0.9 1.2 2.2 3 -5.9 -2.1
dν/dP (cm-1/GPa) 5.5 14 6.4 8.8 7.5 13 4.22 4.16 4.19 4.5 3.8 4.1
Fg-l (pN) 15 15 14 19 33 42 -130 -59
dF/dP (pN/GPa) 82 212 98 137
F6/F7
104 182
0.70 0.70 0.71 0.72 0.68 0.69 0.74
119 112 124 116 94 110
0.75 0.73 0.74 0.71 0.67 0.67 0.65
-2.1 -2 -1.2 -1.9
4.44 4.03 5.92 4.11 4.93
-44 -42 -24 -38
94 85 118 83 91
0.76 0.76 0.72 0.72 0.67
-10 -8 -9 -10.3 -3 -8 -10 -19.8 -17
6.1 5 4.8
23 19 20
11 7.91 10 7.4 -1.8
-38 -31 -37 -41 -12 -31 -40 -78 -67
45 31 40 29 -7
0.72 0.71 0.72 0.74 0.71 0.68 0.69 0.72 0.72
-12.1
6.7
-66
36
0.72
-1436 -241
-57 -109 -104 -76
0.71 0.73 0.73 0.73
-487
-49
0.66
-130
82 164
0.71 0.71
-353 -51
-14 -22.97 -21.96 -16.06
-119
-12
-3 -29 -20.5 -10
1.9 3.78 -9.48
-473 -344 -169
-6.51
-161 -108
0.74 0.76 0.76 0.77
-12.8 -12 -10.7 -8.7
2 2.7 3.2
-211 -203 -182 -141
33 46 54
0.75 0.76 0.78 0.74
-2.2
3.02
-43
59
0.73
-2.8 -1.4 -7
9.6 8 6.2 5.8
-36 -19 -90
124 109 80 72
0.68 0.68 0.68 0.71
-1.9 -1.3 -1.6 -8
3.64 3.99 4.41 1.41
-49 -33 -42 -237
94 103 115 42
0.72 0.72 0.73 0.69
6
35
0.73
-4
1.7
-80
34
0.72
-2
1.3
-60
39
0.73
1092 611 625
0.70 0.71 0.72
19.48 10.07 9.91
3358 J. Phys. Chem. B, Vol. 102, No. 18, 1998 clearly indicates the complexities involved in rigorously treating polyatomic normal modes. Most troublesome is the fact that the measured frequency shift in general reflects a weighted average of the solvation forces along different bonds. Thus inversion of a measured frequency to a force along a particular bond in a polyatomic may not in all cases be rigorously possible. Fortunately, in a number of special cases of practical interest, the force on a given bond may nevertheless be derived even for a polyatomic vibrational mode. For example, vibrations that are more or less localized on a single bond may be reasonably treated as pseudodiatomic vibrations.19,20 Examples of this type include C-X single bonds in which X represent a halogen, oxygen or sulfur atom, or isolated CdC or CdO double bonds embedded in polyatomic molecules. Furthermore, vibrational normal modes involving the symmetric motions of several equivalent bonds, such as the C-C bonds in an aromatic ring breathing mode, C-Cl bonds of carbon tetrachloride, or C-H bonds in a methyl group, contain only one unique bond and thus the corresponding frequency shifts may again be related to the force along that bond.9 Note that asymmetric normal mode vibrations are more problematic as these cannot reasonably be modeled as pseudodiatomic vibrations, unless it is by assuming a complete decoupling of the solvent force along each solute bond (which may well not be a good approximation). Thus eq 6 or eq 7 are only appropriate for those polyatomic normal modes which may be reasonably treated as pseudodiatomic vibrations. In this work, such normal modes are treated as vibrations involving only the two atoms on either side of the bond of interest. The diameters of these atoms are taken to be twice the van der Waals radii of the atoms.21 This method is somewhat simpler than that employed in some previous studies,19,20 in which pseudoatoms diameters were chosen to represent molecular subgroups on either side of the bond (with a centroid corrected bond length). In comparing these two pseudodiatomic models, we find that they produce virtually identical results (within the experimental frequency shift error), and thus the former, simpler of the two approaches, is used. The only parameters required in order to make use of eq 7 are the force constants f and g and the harmonic vibrational frequency νe of the molecule in the reference state (which is sufficiently close to the measured transition frequency, ν0, that for the purposes of this study, the two may be interchanged). In applying eq 6, an additional parameter, representing the effective hard sphere diameter of the solvent,16,17 is required in order to calculate h. Note that forces measured using eqs 6 or 7 are actually differential forces, measured relative to the chosen reference state. Thus, to determine the total solvation induced force on a bond, the parameters f, g, νe (or ν0), and the frequency shift (relative to the reference state) ∆ν must pertain to the isolated (gas phase) reference state. On the other hand, if one is interested in determining the excess force exerted by the solvent relative to the ambient liquid state, then ∆ν must be taken to be the frequency shift relative to the liquid at 1 atm. Furthermore, in principle, f, g, and νe should also pertain to this reference state. In practice, however, the most significant contribution to the force is the frequency shift ∆ν, while solventinduced changes in f, g, and νe rarely contribute more than a few percent to the resulting force. Thus the error produced by approximating νe by the measured vibrational frequency in either the vapor or liquid phase, or by using f and g values derived from experimental normal-mode analysis or quantum calculations for an isolated molecule, are not expected to be significant in comparison with experimental errors in ∆ν (in addition to
Hutchinson and Ben-Amotz
Figure 1. Raman spectra in the CdC stretch (∼1650 cm-1) region of pure liquid octenes (A-C) and 10 wt % solutions of octene in methanol (D-F); (A) 1-octene, (B) trans-2-octene, (C) trans-4-octene, (D) 1-octene/methanol, (E) trans-2-octene/methanol, (F) trans-4-octene/ methanol.
uncertainties associated with approximating a polyatomic normal mode as a pseudodiatomic vibration). IV. Frequency Shift Measurement Procedure The microscope-based Raman system used in these studies has been described elsewhere,22 and thus only modifications to this system are noted here. The excitation source is a 40 mW HeNe laser (Spectra Physics Model 127). Back-scattered signal is collected by an achromatic 5 cm focal length lens and focused into a 200 µm fiber (Nimbus Optics) whose output end is subsequently imaged onto a liquid nitrogen cooled CCD detector (Princeton Instruments LN-CCD) mounted to a home-built lens coupled imaging spectrograph (with a 250 mm focal length). The diffraction grating used in this spectrograph (originally specified for a ISA-320 spectrograph) has 600 gratings/mm and 500 nm blaze wavelength. Exposure times are 10 s per spectrum. For high-pressure studies, a diamond anvil cell (DAC) is used to contain octene and octene/methanol solutions. Ruby fluorescence is used as the internal pressure standard,23-26 derived from the linear shift of the Ruby R1 line with applied pressure, with a slope of 2.74 nm/GPa. Gas-to-liquid frequency shifts are obtained using a coupled TGA/FTIR system. The TGA (TA Instruments Model 951) is coupled via short Teflon gas transfer line to an FTIR (PerkinElmer 1600) that utilizes a 10 cm gas cell with KBr windows. Gas measurements were made at the boiling point of 1-octene (123 °C). For the purposes of the study, samples of the octene family (1-octene, trans-2-octene, and trans-4-octene) and methanol were obtained from the Aldrich Chemical Co. at their highest purity. The samples were used as they were received. Figure 1 shows the CdC stretch band (around 1650-1700 cm-1) in the Raman spectra of the three octenes under ambient conditions, in both the pure liquid and in a methanol solutions (at concentrations of 10 wt %). Frequency shift results for these fluids are collected in Table 1 (and Table 3). Other fluids whose high-pressure frequency shifts were measured in this work are three methyl siloxane derivatives: hexamethyldisiloxane (HMDS), octamethyltrisiloxane (OMTS), and decamethyltetrasiloxane (DMTS). These were handled in
Molecular Force Measurement
Figure 2. Pressure induced frequency shift of the CdC stretch vibration of octenes in the pure liquid (closed symbols) and 10 wt % methanol solutions (open symbols). The frequency shifts are all measured at room temperature (23 °C), relative to the liquid at 1 atm.
J. Phys. Chem. B, Vol. 102, No. 18, 1998 3359
Figure 3. Pressure induced force on the CdC bond in octenes derived from the frequency shifts data shown in Figure 2.
the same way as the octene fluids. The results of these frequency shift measurements are reported in Table 3. V. Results and Discussion Overview of Molecular Force Measurement Results. Molecular forces are determined in this work using two sources of experimental data. One pertains to the transfer of a molecule from the gas to the liquid (or solid) phase. This produces a gas-to-liquid (or gas-to-solid) shift ∆νg-l, which reflects the total force Fg-l, on the solute bond resulting from its interactions with the solvent (at 1 atm). The second type of data comes from pressure induced frequency shifts measured in liquids (or solids) at high pressures (between 0.1 and 10 GPa). Over this pressure range, vibrational frequencies typically shift roughly linearly with pressure and so are well represented by the pressure derivative of the frequency dν/dP, or the corresponding derivative of the force along the bond dF/dP. The experimental data used to measure molecular forces includes both new Raman frequency shift results and a compilation of previously reported gas-to-liquid and high-pressure data. The new measurements performed on 1-octene, trans-2-octene, and trans-4-octene, in both the pure liquid state and as dilute solutions in methanol, illustrate the way in which the magnitude and sign of the solvation force changes as a function of solvent density (pressure) over the gas to high-pressure liquid range. The solvation force is typically elongational (negative) at low density and compressive (positive) at high density (as shown in Figure 4 and discussed below). These results also demonstrate the insensitivity of the solvation force to the location of a bond within a solute and, perhaps more surprisingly, the insensitivity of the solvent-induced force to the molecular structure of the solvent (as shown in Figures 2 and 3 and discussed below). These same general trends are born out by force measurements in a wide variety of other liquid and solid systems (see Table 3). In general, these reveal that bonds of the same type tend to experience similar solvent forces, while bonds of different types experience widely differing gas-to-liquid and pressure-induced solvation forces. The remainder of this section begins with an evaluation of the accuracy and detection limits of forces obtained from vibrational frequency shifts, followed by a more detailed description of the octene CdC shift results, and the general trends emerging from comparisons of molecular forces in different systems.
Figure 4. Density dependence of the frequency shift and force for the CdC bond in 1-octene over the gas to high-pressure liquid density range. Frequency shift and force values pertain to the isolated molecule (gas phase) reference state. The lowest density data point is that in the liquid at 1 atm. The higher density points are at elevated pressures (the same as those in Figures 2 and 3, and Table 1). The curves represent predictions of the perturbed hard fluid (PHF) model with the dashed curves corresponding to a (1% variation in the effective hard sphere diameter of 1-octene (see text for details).
Force Measurement Uncertainty. The precision of good condensed phase vibrational frequency measurements is typically of the order of (0.5 cm-1. Measuring liquid vibrational band positions with an absolute precision of (0.1 cm-1 is extremely difficult (as the corresponding bandwidths are typically of the order of 10 cm-1). It is not at all uncommon to find discrepancies of several cm-1 between vibrational band position for the same fluid, measured in different laboratories. Such uncertainties are clearly illustrated by ASTM Report No. E1840-96, “Establishment of Raman Shift Standards for Spectrometer Calibration”,27 which confirm that the best liquid frequency standards are only reliable to within about (0.5 cm-1. Such frequency measurement inaccuracies place clear bounds on the accuracy of forces derived from frequency shift measurements. When frequency shifts are translated into forces, using eqs 6 or 7, it is clear that the relatiVe accuracy of the force measurements is of the order of (10 pN. The absolute accuracy of force measurements is undoubtedly somewhat lower. This may be inferred from differences between the forces derived using eq 6 and eq 7. The forces in Table 3 are those obtained from eq 6, while the column labeled F6/F7 gives the ratio of the forces obtained using eqs 6 and 7. This ratio is typically of the order of 0.7, implying an absolute force uncertainty of the
3360 J. Phys. Chem. B, Vol. 102, No. 18, 1998 order of 30 pN (assuming that the predominant source of error is the uncertainty in second-order force contribution). Given the approximations made in modeling solute vibrations (and in some cases also the larger errors in the experimental frequency shifts and/or estimated anharmonic force constant values), it is possible that the absolute force measurement uncertainty may be as high as (100 pN. Keeping these uncertainties in mind, it is nevertheless possible to derive some interesting quantitative conclusions and global trends from gas-to-liquid and pressure induced molecular forces measurements. Force on the CdC Bond in Octenes. The results of our measurements of the frequency shift of the CdC stretch in octenes (both in the pure liquids and in methanol solutions) are given in Table 1. The molecular parameters used to analyze these and other previously reported gas-to-liquid and pressureinduced frequency shifts are contained in Table 2. The second column in Table 2 indicates the phase in which the corresponding ν0 at 1 atm were measured (gas, g; liquid, l; or solid, s). The references in the last column of Table 2 are those from which the gas-to-liquid (or solid) and high-pressure results were taken (when two references are given the second pertains to the source of the gas-to-liquid shift data).9,19,20,29-45 The remaining parameters in Table 2 were obtained either from the above references or from standard tables.16,17,21,46 The harmonic and anharmonic force constants f and g (or A ) reg/f) for the CdC and Si-O bonds in the octenes and methyl siloxanes were determined using Gaussian94 ground-state energy calculations,47 scanned as a function of bond length and fit to a cubic anharmonic function. Table 3 contains the corresponding experimental frequency shifts and associated solvation force results. Figure 2 shows the pressure induced frequency shift of the octene CdC vibrational frequencies (relative to the 1 atm liquid state). Two aspects of these results are particularly noteworthy. First of all, the frequency shifts do not depend on the location of the CdC bond along the octene backbone. This behavior supports the pseudodiatomic approximation made in deriving eqs 6 and 7, as the measured solvation force is independent of the lengths of the chains attached to either end of the CdC bond. The second striking feature of these results is the insensitivity of the frequency shifts to the molecular structure of the solvent (which is either octene for the pure liquids or methanol for the solutions). Figure 3 shows the octene frequency shift results translated into a pressure-induced force along the CdC bond (using eq 6). These results indicate that, in keeping with the corresponding frequency shifts, the force is virtually independent of the location of the CdC bond or the molecular structure of the solvent. The slightly larger spread in the force as opposed to the frequency shift results (shown in Figure 2) may reasonably be attributed to uncertainties in the parameters (in eq 6) used to translate the frequency shifts into a forces. To be more specific, although it is not impossible that there is a slight dependence of force on the location of the CdC bond within the octene chain, these are too small relative to the combined errors in the frequency shift measurements and parameter estimates to be reliably determined from these data. Note that the frequency shifts and forces shown in Figures 2 and 3 pertain to the ambient liquid reference state. This means that the forces in Figure 3 represent the excess solvation force on the CdC bond relatiVe to that in the ambient liquid rather than the total solvation force (relative to the isolated solute). Figure 4, on the other hand, shows the total frequency shift and solvation force for 1-octene referenced to 1-octene in the
Hutchinson and Ben-Amotz gas phase. This clearly reveals a more interesting nonmonotonic dependence of the force on solvent density. The initial gasto-liquid shift and corresponding solvation force are negative, which means that the CdC bond experiences an elongational force when it is transferred from the gas to the ambient liquid. As the density is further increased (at high pressure), the CdC frequency begins to climb back up toward the gas-phase value (∆νg-l ) 0) and eventually to rise above the gas-phase frequency at the highest pressures. In the high-pressure region, the force is a linear function of pressure (as shown in Figure 3), although it is a nonlinear function of density (as shown in Figure 4). The curves in Figure 4 are derived from predictions of the perturbed hard fluid (PHF) model9,19,20,28 using parameters given in Tables 1 and 2. The dashed curves represent PHF predictions using solvent diameters which are either 1% larger or smaller than that in Table 2. Although such PHF modeling of frequency shifts is not the main focus of this work, it is instructive to note that, in the course of the PHF analysis, the frequency shift is separated into repulsive contribution arising from hard-core collisions between the solute and solvent, and an attractive contribution arising from long range cohesive (dispersion and multipolar) interactions. The attractive mean field contribution is predicted to have a linear density dependence while the repulsive contribution has a strongly nonlinear density dependence. It is the balance of these two opposing solvation forces that produces the change in sign of the frequency shift and force with increasing density. The PHF analysis also yields a parameter corresponding to the density derivative of the attractive frequency shift of (CA ) -0.99 cm-1/nm-3 for 1-octene in the pure liquid) which represents an attractive contribution to the CdC bond solvation force in 1-octene of FA(pN) ≈ -28F, where F is the density of 1-octene in units of molecules/nm3. Global Trends in Solvent-Induced Forces. The results in Table 3 reveal that different types of bonds undergo characteristic gas-to-liquid and high-pressure frequency shift and solvation force behavior. For example, C-C stretches typically have a small positive gas-to-liquid shift. This behavior is characteristic of C-C single bonds, as most other bonds have negative gas-to-liquid shifts of larger magnitude. The pressure shift of C-C stretches vary from 100 to 200 pN/GPa. Other carbon-carbon vibrations (CdC and aromatic ring breathing modes) have very similar pressure shifts to those of C-C single bonds. This makes physical sense because the repulsive forces experienced by carbon-carbon bonds might be expected to depend primarily on the size of the carbon atoms and not on the carbon-carbon bond order. On the other hand, unlike the C-C single bonds, double-bonded and aromatic carbon stretches typically have large negative gas-to-liquid shifts and thus negative solvation forces of the order of -50 pN. This behavior illustrates the importance of intermolecular attractive forces in the solvation for π-bonded vibrational modes. In other words, π-bonded atoms typically experience an elongational solvation force at ambient liquid densities (implying that the excited vibrational state is better solvated than the ground vibrational state). The C-H (and C-D) stretch modes experience dramatically different solvation force behavior. These bonds typically have large negative gas-to-liquid shifts and very small (usually positive) high-pressure shifts. The gas-to-liquid forces on C-H bonds are roughly -50 pN, again indicating an attractive (elongational) solvation force. At high pressure, C-H solvation forces increase with a very small slope of 20-30 pN/GPa. This relatively weak pressure dependence is consistent with the
Molecular Force Measurement smaller size of the hydrogen atom (which has a smaller crosssectional area of exposure to the solvent than the carbon in a C-C bond). Hydroxyl, O-H, (and amine, N-H) vibrations, on the other hand, have very large negative gas-to-liquid and pressure shifts (even in nonpolar solvents). The large negative gas-to-liquid forces, which are of the order of several hundred to -1000 pN, are consistent with the strongly attractive solvation of these polar bonds (solvation favors elongation and softening of the O-H bond). The high-pressure forces on O-H bonds, unlike those on C-H bonds, are typically negative with a pressure derivative of the order of -100 pN/GPa. This behavior indicates that the attractive interaction strength of O-H bonds continues to grow with increasing density even at very high pressures (thus the O-H bond further elongates and softens with increasing pressure). Previous studies suggest that hydrogen vibrations (both C-H and O-H), unlike those of other bonds, experience a nonlinearly density dependent attractive frequency shift.13,20 This has tentatively been attributed to hydrogen bond formation (even with C-H bonds) whose strength increases nonlinearly with density.20 Although the present results cannot resolve the source of the force along hydrogen containing bonds, the large gasto-liquid and negative high-pressure shifts (of O-H vibrations) confirm the softening of hydrogen vibrations in high-density fluids. The C-O vibrations have moderately large negative gas-toliquid shifts and positive high-pressure shifts. The negative gasto-liquid force is of the order of -100 pN, and the pressure induced force is positive with a magnitude of about 100 pN. Note that pressure shift is similar to that for the carbon-carbon vibrations (in keeping with the similar size of O and C), while the gas-to-liquid force is negative, again indicating the sensitivity of gas-to-liquid shifts to the nature of the bond, and the softening of the C-O bond upon solvation. The carbonyl CdO bonds experience much stronger attractive solvation forces than single C-O bonds. The gas-to-liquid force on carbonyls are nearly as large as those of O-H bonds. Most strikingly, as in the case of the O-H vibrations, the attractive (elongational) force persists even at very high pressures, as reflected in the large negative pressure derivatives of between -100 and -200 pN/GPa. Thus, although the same solute atoms are involved (C and O), the pressure shift of the carbonyls is opposite in sign to that of the C-O. This indicates that, unlike the π-bonded carbons (which experience the same pressure shift as single C-C bonds), the polarity of the carbonyls induces an attractive solvation force that is much larger in magnitude and persists to higher pressures than that of nonpolar π-bonds (i.e. alkene and aromatic CC bonds). The negative sign of the pressure-induced force (such as that on O-H bonds) reflects that fact that the frequency shift in these compounds has not yet reached its minimum value even at pressures of several thousand atmospheres. At higher pressures, the force on carbonyls is expected to become positive (and of similar magnitude to that on C-O and C-C bonds), although most carbonyl containing fluids freeze before this happens. Nitrile, CtN, vibrations have negative gas-to-liquid shifts that are nearly as large as those of carbonyls, while their pressure shifts are significantly more positive. The gas to-liquid force on nitriles is typically near -200 pN, and the pressure induced force is close to 50 pN/GPa. The small magnitude of this pressure induced force again suggests that over the pressure range of the experiments (between 0.1 and 1 GPa) the nitrile frequency shift is near its largest negative (gas-to-liquid) shift
J. Phys. Chem. B, Vol. 102, No. 18, 1998 3361 and is only beginning to turn upward. Thus (as for the CdO and O-H bonds) the pressure shifts of the nitriles reflect a balance of attractive and repulsive solvation forces which in this case only mildly favors repulsion (while in the case of carbonyls and hydroxyl groups attractive solvation continues to dominate even at such high pressures). The nitrogen molecule, NtN, experiences a smaller negative gas-to-liquid force and a slightly larger pressure induced force than CtN. This again suggests the role of polarity in enhancing attractive solvation forces (since the NtN and CtN bonds are otherwise identical in bond order and number of π electrons). Vibrations involving iodine, C-I and I-I, undergo generally similar gas-to-liquid and pressure shifts (with the exception of solid I2). The gas-to-liquid forces on the X-I vibrations are typically near -50 pN, while the corresponding pressure derivatives are near 100 pN/GPa. The markedly different behavior in solid I2 (which has a much larger negative gas-tosolid shift and a smaller positive pressure derivative) suggests a significantly different coupling between the external pressure and I2 in liquid and crystalline molecular environments. By far the largest pressure-induced forces found for any bond are those on the Si-O stretch in methyl siloxanes (hexamethyldisiloxane, octamethyltrisiloxane, and decamethyltetrasiloxane). These forces are of the order of 1000 pN/GPa (1 nN/ GPa), or about a factor of 10 larger than pressure-induced forces of most other bonds. The reason for this cannot simply be contributed to the larger size of the Si atom, as this is not enough to explain such a large force increase. The explanation may lie in the fact the each Si in these compounds is saturated with methyl groups. Thus it appears that these methyl groups serve to significantly increase the effective area over which the solvent pressure is transmitted to the Si-O bond and thus to produce a correspondingly large pressure induced force. VI. Conclusions A method for experimentally measuring forces experienced by molecules using shifts in their vibrational frequencies has been demonstrated. This relies on a predicted proportionality of the measured frequency shift and the mean force along the vibrating bond, with a coefficient that depends only on properties of the isolated solute (and the solvent diameter required in approximating the second-order force contribution to the frequency shift). The forces measured using this method are estimated to have an experimental precision of the order of (10 pN (with an absolute uncertainty which may be as large as (100 pN). Measurements of gas-to-liquid and pressure-induced frequency shifts yield forces relative to the isolated molecule and ambient liquid reference state, respectively. These forces are typically negative for the former process and positive for the latter. Physically this implies that upon solvation molecules usually experience an elongational force, while at high pressure they experience a compressional force. This general behavior clearly illustrates a delicate balance of forces present in a condensed phase, and quantitatively measured using this technique. The results reveal marked differences between the forces experiences by different types of bonds, with weak dependence on the location of the bond within a molecule or the nature of the solvent. In addition, gas-to-liquid forces are found to be more sensitive to solute bond order and polarity than pressure induced forces, which appear to be sensitive primarily to the effective size of the atoms (or atomic groups) on either side of the vibrating bond.
3362 J. Phys. Chem. B, Vol. 102, No. 18, 1998 The largest gas-to-liquid forces are found for the O-H bond in methanol (which are over -1000 nN). Nearly as large solvation forces are found for O-H bonds dissolved in nonpolar solvents as well as for CdO and CtN bonds (with forces approaching -500 nN). Interestingly, O-H bonds are found to soften at high pressure not only in hydrogen bonding solvents but also in CS2 solutions. Solvation induced forces typically depend linearly on pressure (and are often of the order of 100 nN/GPa), as anticipated by simple mechanical arguments for a solute dissolved in a continuum solvent. The effective bond cross-sectional areas derived from the measured forces are of the same order as those estimated from the cross-sectional areas of the atoms on either side of the vibrating bond. The only marked exception is for Si-O bonds in methyl siloxanes, for which pressure-induced force are about 10 times larger than those of other bonds. The reason for this may be the larger effective cross section produced by the methyl groups surrounding each Si atom. Finally, pressure-induced forces in the solid phase are found to be of the same order as those in liquids, although there is some evidence that these may differ to a measurable degree when comparing the liquid and solid phase of the same material. Acknowledgment. This work was supported by the Office of Naval Research (Grant N00014-95-1-0403). We also thank Prof. Wei-Ping Pan at the Department of Chemistry, Western Kentucky University, for his help in obtaining the gas-to-liquid frequency shift of 1-octene. References and Notes (1) Carpick, R. W.; Salmeron, M. Chem. ReV. 1997, 97, 1163; Jarvis, S. P.; Tokumoto, H. Probe Microsc. 1997, 1, 65. (2) Odell, J. A.; Keller, A.; Rabin, Y. J. Chem. Phys. 1988, 88, 4022. (3) Odell, J. A.; Muller, A. J.; Narh, K. A.; Keller, A. Macromolecules 1990, 23, 3092. (4) Odell, J. A.; Keller, A. J. Polym. Sci., Part B: Polym. Phys. 1986, 24, 1889. (5) de Souza, L. E. S.; Guerin, C. B. E.; Ben-Amotz, D.; Szleifer, I. J. Chem. Phys. 1993, 99, 9954. (6) Buckingham, A. D. Proc. R. Soc. London, Ser. A 1958, 248, 169. (7) Buckingham, A. D. Proc. R. Soc. London, Ser. A. 1960, 255, 32. (8) Buckingham, A. D. Trans. Faraday Soc. 1960, 56, 753. (9) (a) Ben-Amotz, D.; Herschbach, D. R. J. Phys. Chem. 1993, 97, 2295. (b) Ben-Amotz, D. J. Phys. Chem. 1993, 97, 2314. (10) Ravi, R.; de Sousa, L. E. S.; Ben-Amotz, D. J. Phys. Chem. 1993, 97, 11835. (11) de Souza, L. E. S.; Ben-Amotz, D. J. Chem. Phys. 1994, 101, 4117. (12) Oxtoby, D. J. Chem. Phys. 1979, 70, 2605. (13) Zakin, M. R.; Herschbach, D. R. J. Chem. Phys. 1988, 89, 2380. (14) Fischer, S. F.; Laubereau, A. Chem. Phys. Lett. 1975, 35, 6.
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