The search for the hydrophobic force law

Abstract

After nearly 30 years of research on the hydrophobic interaction, the search for the hydrophobic force law is still continuing. Indeed, there are more questions than answers, and the experimental data are often quite different for nominally similar conditions, as well as, apparently, for nano-, micro-, and macroscopic surfaces. This has led to the conclusion that the experimentally observed force-distance relationships are either a combination of different 'fundamental' interactions, or that the hydrophobic force-law, if there is one, is complex--depending on numerous parameters. The only unexpectedly strong attractive force measured in all experiments so far has a range of D approximately 100-200 angstroms, increasing roughly exponentially down to approximately 10-20 angstroms and then more steeply down to adhesive contact at D = 0 or, for power-law potentials, effectively at D approximately 2 angstroms. The measured forces in this regime (100-200 angstroms) and especially the adhesive forces are much stronger, and have a different distance-dependence from the continuum VDW force (Lifshitz theory) for non-conducting dielectric media. We suggest a three-regime force-law for the forces observed between hydrophobic surfaces: In the first, from 100-200 angstroms to thousands of angstroms, the dominating force is created by complementary electrostatic domains or patches on the apposing surfaces and/or bridging vapour cavities; a 'pure' but still not well-understood 'long-range hydrophobic force' dominates the second regime from approximately 150 to approximately 15 angstroms, possibly due to an enhanced Hamaker constant associated with the 'proton-hopping' polarizability of water; while below approximately 10-15 anstroms to contact there is another 'pure short-range hydrophobic force' related to water structuring effects associated with surface-induced changes in the orientation and/or density of water molecules and H-bonds at the water-hydrophobic interface. We present recent SFA and other experimental results, as well as a simplified model for water based on a spherically-symmetric potential that is able to capture some basic features of hydrophobic association. Such a model may be useful for theoretical studies of the HI over the broad range of scales observed in SFA experiments.

Fig. 1

Results of experimentally measured forces between hydrophobic surfaces under different conditions presented as a semi-log (A) and a log-log (B) plot are indicated as followed: red squares: DMDOA, LB-deposited, deaerated, Wood and Sharma; green triangles: OTE, chemical vapor deposition, deaereted; all data points in blue were obtained by Meyer et al., using OTE and DODA surfaces prepared by LB-depostion. Suggested regimes are marked as the short range, long range, and ES/BC, and are discussed below. The shaded VDW force band corresponds to Hamaker constants between 3 and 10 × 10⁻²¹ J.

Fig. 2

The potential of mean-force, PMF, in terms of the radial distance, r, between the pair of hydrophobes, is given above. The solid curve is obtained through our Monte Carlo simulations utilizing the spherically-symmetric water model. The dotted curves roughly depict the reported results of Ludemann et al. and Shimizu et al.; these two employ conventional semi-empirical force-fields for water, SPC and TIP4P, respectively. Our PMF has a slightly higher frequency and a moderately lower amplitude. The curves are shifted up–down so as to fix the “barrier” at zero energy.

Fig. 3

The potential of mean-force, PMF, in terms of the radial distance, r, between the pair of hydrophobes, is given above for the variable (A) and constant (B) scenarios. Each curve represents a different temperature. The variable case but not the constant case notably exhibits the following trend: with increasing temperature, the wells deepen, especially the one corresponding to binding. The curves are shifted up–down so as to fix the “barrier” at zero energy.