Crystallization --> General

Thermodynamic potential of crystal growth.

by Leonid Sakharov

A process of phase transition of first-order such as crystallization is accompanied with heat release that reflected a difference between energy levels of crystal phase and surrounding it matter that is the 'soil' for the growing crystal. The amount of heat is equivalent to difference of enthalpies between characterized crystalline phase and in initial phase :

Qcryst =-ΔH = Hcryst - Hraw  (1),

where ΔH - change of enthalpy during crystal growth,  Hcryst and Hraw - enthalpies of crystal phase and initial feeding phase correspondently.

Depend on temperature crystalline area should expand itself or shrink down into raw phase. The criteria for the process direction gives a value of free energy potential. If it is negative a crystal will grow up if positive crystal will dissolve. For most practical applications the Gibbs free energy represents thermodynamic potential when temperature and pressure is constant along system. The typical situation of material in crucible in high temperature furnace. The difference of Gibbs potentials in crystalline phase and feeding phase (liquid, gas or plasma) can be represented by formula:

ΔG = ΔH - ΔS*T   (2),

where ΔS =  Scryst - Sraw, change of entropies between crystalline and liquid (from this point specificity sake we will talk about liquid as an initial phase for crystal growth), T - absolute temperature in the system (actually important that it will be temperature at the border between phases).

Entropy is a measure of disorder in the system. A crystalline phase is characterized with  long-range order in atoms locations and entropy in it lower then in liquid. The value of changing entropy ΔS is negative the same as value of changing enthalpy ΔH for process of transformation of liquid into crystal crystal.

Changing of enthalpy ΔS can be measured indirectly if heat of crystallization and a temperature of equilibrium, known also as melting point, is defined experimentally by calorimetry methods:

 ΔS = ΔH (3) , To

where To - melting point, temperature equilibrium temperature when growth rate is effectively zero. Changing of Gibbs free energy in this case can be presented in form:

 ΔG = ΔH* ΔT (4) , To

where ΔT = To - T , is often called supercooling. Note that if temperature less than equilibrium and keeping in mind that ΔH, the change of enthalpy, is negative too;  ΔG change of Gibbs thermodynamic potential is less than zero and crystal will grow up. Slightly confusing but can be handling all right just remembering that there is some most important characteristic for crystal growth namely melting point. The more temperature is deviate down from melting point  the large thermodynamic force of crystallization.

The discussion above was limited to the case when chemical composition of noncrystalline phase is the same as for growing crystal. Like it for the case of distillate water for example. In real live it is never the case as soon the are always impurity additions in any material. Formula (2) has to be amended by changing of chemical potential in crystal phase and initial noncrystalline phase:

ΔG = ΔH - ΔS*T + k*T*ln(Ccr)           (5),

where k - Boltzmann's constant  (1.380 6504(24) × 10−23 J K−1 ), Ccr - effective concentration of crystal elements in initial phase, ln() is function of natural logarithm.  The term an effective concentration of crystal elements demands clarification. In some sense it is quite straightforward. Let's do imaginary experiment cooling down water with salt dissolved in it. Maximum amount of ice that can be formed too initial amount salt water is represent the concentration in formula (5). The tricky part is that amount should be measured in molecules that is not always obviously defined number. Let say the sample is made from two organic compounds both of them stable and do not react up to temperature of ideal mixing with each other. In this case there is possibility to directly apply molar concentration of components in mixture to formula (5) for each of them. But variety of situation could be very broad especially in case when for compounds like for salts there is no way to isolate molecule as building block of crystal structure. There is for example possibility when more then one crystal with different compositions but these include the same atoms at least part of them. In this case there is the way experimentally define an effective concentration for each of crystals from defining temperature of liquidus Tl - temperature of equilibrium for crystal with initial phase with given concentration Ccr. In this situation equilibrium (5), taking into account expression (3), can be transformed into:

 ln(Ccr) = ΔH ( 1 - 1 ) (6) k To Tl

Note again thatas soon ΔH has negative value liquidus has to be less or equal then melting temperature:  Tl <= To. If in formula (5) make substitution values changing entropy with defined in formula (3) and  ln(Ccr) from formula (6) the value of changing free energy of crystallization can be defined analogously like in formula (4) for crystal growth from liquid with the same chemical composition as crystal with only difference that melting temperature is replaced with liquidus temperature:

 ΔG = ΔH* ΔT* (7) , Tl

where ΔT* = Tl - T .

Equation (6) has very practical sense making possible experimentally estimate melting temperature and enthalpy of crystallization even for situation when crystal of the material is not stable at melting point. A zinc oxide presents ideal example of application of indirect measurement of melting point temperature and enthalpy of crystallization on the date of data for liquidus for series of sample with different concentrations.

According to reference-books ZnO has melting point at 1975oC that is actually temperature of decomposition. But the patent USA 3,043,671 "Zink oxide crystal growth method" provides chart with data of liquidus temperatures of ZnO in eutectic solution with PbF2. The software LeoStatistic was used to extract data from chart

in numerical form. After converting temperature  Celsius (C) degrees into absolute Kelvin scale the approximation with equation:

ln(c) = a + b/T      (8)

will be done:

The result approximation formula is ln(C) = 1.699 -4238*1/T

Coefficients in formula (8) are a = 1.699 and b=  4238. These values are used to calculate melting point and enthalpy of crystallization:

To = b/a = 4238/1.699 = 2494 K = 2221oC

ΔH = 1.38×10−23 JK-1 × 4.238×103 K = 5.84×10-20 J per one molecule

and  molar enthalpy of crystallization:

ΔHm = ΔH × Na  = 5.84×10-20 J × 6.02 ×1023 mol-1 = 35.2 kJ/mol

Submited at Mar. 4, 2010; 17:24