The attachment of solid spherical particles to one another under dynamic conditions has been investigated theoretically. The particles are assumed to be uniform with regard to their bulk properties; however, their surfaces have been altered so that one particle is hydrophobic, another hydrophilic. The role of modification has been revealed in the "apparent" slippage of liquid over a solid hydrophobic surface, and in an increase in the critical interlayer thickness at which the particles coagulate. A problem similar to the Reynolds one has been solved for the purpose of determining the force of hydrodynamic resistance to the approach of particles to one another (the difference between the problems consists only in the change of boundary conditions on the hydrophobic surface, and in the spherical shape of interacting bodies). It has been shown that in the case under consideration the resistance to the motion cannot be determined by the Taylor formula. This formula later must be multiplied by a correction function dependent only on the ratio of the quadruple slippage coefficient to the interlayer thickness. The equation for the approach of bodies to one another under the effect of the hydrodynamic resistance force has been solved. An expression has been derived for the particles approach velocity. On its basis, criteria for the attachment of particles to one another have been formulated in the form of a condition for the Reynolds number. It has been shown that in limit cases a hydrophobic particle behaves like a hydrophilic one, or moves like a bubble. An intermediate case can also be realized, in which the motion velocity necessary for coagulation depends on the slippage coefficient value.