494 J. TF. Gibbs — Equilibrhun of Heteroyeneons /Siibstances. 



But even when tlie fluid is supersaturated only so much as is 

 necessary in order that the crystal shall grow at all, it is not to be 

 expected that the form in which 2{o's) has a niiniraum value (or 

 such a modification of that form as may be due to gravity or to the 

 influence of the body supporting the crystal) will always be the 

 ultimate result. For we cannot imagine a body of the internal 

 structure and external form of a crystal to grow or dissolve by an 

 entirely continuous process, or by a process in the same sense continu- 

 ous as condensation or evaporation between a liquid and gas, or the 

 corresponding processes between an amorphous solid and a fluid. 

 The process is rather to be regarded as periodic, and the formula 

 (664) cannot properly represent the true value of the quantities 

 intended unless SJV^ is equal to the distance between two successive 

 layers of molecules in the crystal, or a multiple of that distance. 

 Since this can hardly be treated as an infinitesimal, we can only con- 

 clude with certainty that sensible changes cannot take place for 

 which the exjjression (664) would have a positive value.* 



* That it is necessary that certain relations shall be precisely satisfied in order that 

 equilibrium may subsist between a liquid and gas with respect to evaporation, is 

 explained (see Clausius " Ueber die Art der Bewegung, welche wir "Warme nennen," 

 Poyy. Ann., Bd. c, S. 353 ; or Abhand. iiber die viech. Wdrmethemie, XIV,) by suppos- 

 ing that a passage of individual molecules from the one mass to the other is continually 

 taking place, so that the slightest circumstance may give the preponderance to the 

 passage of matter in either direction. The same supposition may be applied, at least 

 in many cases, to the equilibrium between amorphous solids and fluids. Also in the 

 case of crystals in equilibrium with fluids, there may be a passage of indiAddual mole- 

 cules from one mass to the other, so as to cause insensible fluctuations in the mass of 

 the solid. If these fluctuations are such as to cause the occasional deposit or removal 

 of a whole layer of particles, the least cause would be sufficient to make the probability 

 of one kind of change prevail over that of the other, and it would be necessary for 

 equilibrium that the theoretical conditions deduced above should be precisely satisfied. 

 But this supposition seems quite improbable, except with respect to a very small side. 



The following view of tlie molecular state of a crystal when in equilibrium with 

 respect to growth or dissolution appears as probable as any. Since the molecules at 

 the corners and edges of a perfect crj-stal would be less firmly held in their places 

 than those in the middle of a side, we may suppose that when the condition of 

 theoretical equilibrium (065) is satisfied several of the outermost layers of molecules 

 on eacli side of the crystal are incomplete toward the edges. The boundaries of these 

 imperfect layers probably fluctuate, as individual molecules attach themselves to the 

 crystal or detach themselves, but not so that a layer is entirely removed (on any side 

 of considerable size), to be restored again simply by the irregularities of the motions 

 of the individual molecules. Single molecules or small groups of molecules may 

 indeed attach themselves to the side of the crystal but they will speedily be dislodged, 

 and if any molecules are thrown out from the middle of a surface, these deficiencies 



