Solvent effect on the roughening transition and wetting of n-paraffin crystals

Abstract

The solubility parameters, roughening temperatures and edge free energies have been measured as a function of temperature and interpreted for n-C₂₃H₄₈ and n-C₂₅H₅₂ crystals. The crystals have been grown from solutions consisting of n-hexane and toluene mixtures. The dissolution enthalpy is used to determine the strongest bond in the bulk of the crystal Φ_str^bulk and the roughening temperature is used to determine the strongest bond in the crystalline interface Φ_str^int. The ratio of these two bonds is called the wetting κ of the crystalline interface. It turns out that the solutions containing mixtures of n-hexane and toluene behave according to a regular solution, which follows directly from thermodynamics with the use of a mean field approach. The strongest bond in the bulk of the crystal has the same dependence on the composition of the solution as the dissolution enthalpy. In contrast with this, the strongest bond in the interface stays approximately constant and, therefore, the dependence of the wetting κ on the composition of the solution follows that of the enthalpy of dissolution.

Introduction

The crystallisation of n-paraffins has been studied for many years, because they are interesting model compounds for understanding different crystallisation phenomena, such as two-dimensional nucleation, spiral growth mechanisms, morphology studies, habit control, etc. Furthermore, the crystallisation of n-paraffins is very important to the petroleum industry. Crude oil and its products consists of a significant amount of n-paraffins, which upon cooling will crystallise, starting with the longer chain homologues. At low temperatures this often results in blocking of fuel filters, gelling of crude oil and the formation of slack wax from waxy raffinates. Therefore, a good understanding of the crystallisation of n-paraffins and the influence of external factors such as solvents and additives is necessary. This paper reports the roughening transitions, the roughening temperatures and the wetting of the {1 1 0}/{1 1 1} faces of orthorhombic n-paraffin crystals, crystallising from binary and ternary mixtures.

Burton, Cabrera and Frank were the first to introduce the concept of the roughening of crystal faces [1]using Onsagers theory [2]of order–disorder phase transitions of a rectangular two-dimensional layer with different (or equal) interactions in two directions. In this model the Ising temperature T^c or dimensionless Ising temperature θ^c is characterised by the temperature where the step free energy of a face {h k l}γ_step, which decreases linearly with increasing T, becomes zero. Since that, the roughening transition of crystal faces have been studied in great detail using, among others, Monte–Carlo simulations 3, 4, 5, 6, 7, 8. The roughening transition of a solid-on-solid (SOS) simple cubic and a SOS body centred cubic model is understood to be a continuous phase transition of infinite order. These transitions are characterised by a critical dimensionless temperature θ^R_{h k l}, so that

$$\text{if}\text{θ<θ}{}^{\mathrm{<mtext>R</mtext>}}\text{,}\text{γ}{}_{\mathrm{<mtext>step</mtext>}}\text{>0}$$$$\text{and}\text{if}\text{θ⩾θ}{}^{\mathrm{<mtext>R</mtext>}}\text{,}\text{γ}{}_{\mathrm{<mtext>step</mtext>}}\text{=0.}$$

If the bonds used to calculate the critical temperatures have been scaled relative to an arbitrary reference bond, which is usually the strongest bond Φ_str, also the critical dimensionless temperature has been scaled according to this bond:

$$\text{θ}{}^{\mathrm{<mtext>R</mtext>}}{}_{\mathrm{<mtext>\{h</mtext><mspace\; sp="0.25"\; width="2px"\; linebreak="nobreak"\; is="true"></mspace><mtext>k</mtext><mspace\; sp="0.25"\; width="2px"\; linebreak="nobreak"\; is="true"></mspace><mtext>l\}</mtext>}}\text{=}\text{2kT}{}^{\mathrm{<mtext>R</mtext>}}{}_{\mathrm{<mtext>\{h</mtext><mspace\; sp="0.25"\; width="2px"\; linebreak="nobreak"\; is="true"></mspace><mtext>k</mtext><mspace\; sp="0.25"\; width="2px"\; linebreak="nobreak"\; is="true"></mspace><mtext>l\}</mtext>}}\text{Φ}{}_{\mathrm{<mtext>str</mtext>}}\text{,}$$

where k is the Boltzmann constant, T^R_{h k l} the absolute roughening temperature and Φ_str the energy of the strongest bond in the structure.

Later it was shown that γ_step decreases continuously as a function of the temperature following

$$\text{γ}{}_{\mathrm{<mtext>step</mtext>}}\text{∼}\text{exp}\text{[−α(T}{}^{\mathrm{<mtext>R</mtext>}}\text{−T)}{}^{\mathrm{-1/2}}\text{]T⩽T}{}^{\mathrm{<mtext>R</mtext>}}\text{,}$$

where α is a constant depending on the system. This transition is a so-called Kosterlitz–Thouless transition 9, 10, 11, 12, 13. This behaviour of γ_step has been confirmed by experiments on He crystals and by experiments on n-paraffin crystals grown from toluene solutions 14, 15, 16, 17.

For real crystal faces it is often very difficult to calculate the exact value of θ^R, therefore the dimensionless critical Ising temperature θ^c is calculated instead and it is assumed that this is almost equal to the real roughening temperature θ^R. To do so, a crystal face is characterised by a one-layer interface model where a flat crystal face is divided into cells which are considered to be either in a solid state or in a fluid state. Below this layer, only solid cells exist and above this layer only fluid cells exist. According to statistical mechanics, this one-layer model has an order–disorder phase transition at the temperature T^c (or θ^c) [2]. Below θ^c, the layer consists of two separated phases, namely a phase with a large fraction of solid cells and a small fraction of fluid cells and a phase with a large fraction of fluid cells and a small fraction of solid cells. Above θ^c only one mixed layer with solid and fluid cells exists. For three types of layers (or nets) it is possible to calculate θ^c exactly, that is for a trigonal net where three bonds originate from one node, for a rectangular net where four bonds originate from one node and for an hexagonal net where six bonds originate from one node [18]. Rijpkema and Knops [19]developed a method to calculate θ^c for crystal faces with different structures, which may be the so-called complex connected nets. A review on the determination of the roughening temperature of the crystals faces has been given by Bennema and van der Eerden 18, 20.

The roughening temperature of a face {h k l} will be determined, among others, by the structure of that face together with the anisotropy and the energy of the bonds 8, 21, 22, 23, 24. It turns out [24]that the energy of the bonds is of more importance for the roughening temperature than the anisotropy and the exact structure of the bonds. Only a very high anisotropy has a noticeable influence on θ^c. Using the experimental method of Elwenspoek and van der Eerden, several roughening transitions of {1 1 0}/{1 1 1} faces of orthorhombic n-paraffin crystals grown from different solutions have already been determined 16, 17, 25. In these papers it was assumed that the investigated faces were {1 1 0} faces. Recently, however, Grimbergen et al. [26]and van Hoof et al. [27]showed that the investigated faces can be either {1 1 1} or {1 1 0} faces or both. It has been shown that the nature of the phase transition of such systems depends on the solvent used for crystallisation 17, 25. Solvents, which are alike the molecules in the solid–fluid interface of the crystal which is studied, induce a first-order phase transition, whereas solvents that do not resemble the molecules in the solid–fluid interface induce a Kosterlitz–Thouless transition. The unexpected first-order roughening transition of the {1 1 0}/{1 1 1} faces of e.g. n-C₂₃H₄₈ and n-C₂₅H₅₂ crystals grown from n-hexane solutions has been explained in terms of a surface structural phase transition to a phase related to the well-known rotator phase [28], which causes a sudden decrease of the energy of the bonds in the surface and thus of its roughening temperature 16, 17, 25.

To characterise the nature of a roughening transition, different methods have been proposed which determine the step (edge) free energy at temperatures close to T^R 10, 15, 29, 30, 31, 32. A detailed study and review of these critical phenomena is presented in a forthcoming paper [33]. In this paper the so-called kinetic roughening of a crystal face is used to determine the step free energy [32]. Kinetic roughening is the roughening of a flat face, at temperatures below its roughening temperature (T⩽T^R), caused by the driving force for crystallisation. This driving force is characterised by the supersaturation σ, where

$$\text{σ=}\text{Δ}\text{μ}\text{kT}\text{.}$$

Here Δμ is defined as the difference in chemical potential between a solid and a fluid particle. At supersaturations below a certain critical value σ^c, a nucleation barrier exists causing a surface to grow via a spiral growth mechanism or via a two-dimensional nucleation mechanism (such as “birth and spread”). Both mechanisms result in a flat surface. At supersaturations above σ^c, the nucleation barrier vanishes and so every two dimensional nucleus, regardless of its size will be stable and be able to grow out. This will result in the random addition of stable two-dimensional nuclei, which causes a face to become rough. Elwenspoek and van der Eerden [32]showed that if the 2D nucleation barrier ΔG* vanishes at kT, where ΔG* is the Gibbs free energy of the critical nucleus, the edge free energy can be determined using the critical supersaturation

$$\text{σ}{}^{\mathrm{<mtext>c</mtext>}}\text{=}\text{π}\text{f}{}_0\text{γ}{}_{\mathrm{<mtext>step</mtext>}}\text{ξ}{}^2\text{.}$$

Here f₀ is the surface area per growth unit and ξ is a shape factor determined by the shape of the nucleus. This critical supersaturation σ^c can be determined by measuring the growth rate as a function of the supersaturation and by observing the crystal face at different supersaturations. In the first method, σ^c is marked by the supersaturation where the growth rate curve changes from nonlinear, the growth proceeds via a birth and spread mechanism or via a spiral growth mechanism, to linear, that is the growth proceeds via a rough growth mechanism. The second method marks σ^c by the supersaturation, where a flat face becomes round. Both methods have been used to determine edge free energies for different crystal faces 16, 17, 32, 34, 35. Eq. (5)should be used with care, because the assumption that the nucleation barrier ΔG* vanishes at exactly kT is questionable, because kinetic roughening is not a transition occurring at a critical point but a gradual process, and moreover, the supersaturation where a face becomes macroscopically rough does not have to coincide with the supersaturation where it becomes microscopically rough [33]. Therefore, the value of γ_step which is determined by this method can have an error. However, if the same procedure is used for every experiment (temperature where γ(T) is determined) it will still be possible to determine the nature of roughening transitions, because the relative error in the edge free energies of the same system will be small.

Within the framework of cell models the bond energy Φ is defined by

$$\text{Φ}{}_{\mathrm{i}}\text{=φ}{}^{\mathrm{<mtext>sf</mtext>}}{}_{\mathrm{i}}\text{−}\text{1}\text{2}\text{(φ}{}^{\mathrm{<mtext>ss</mtext>}}{}_{\mathrm{i}}\text{+φ}{}^{\mathrm{<mtext>ff</mtext>}}{}_{\mathrm{i}}\text{),}$$

where i is the ith bond type, φ_i^sf the bond between an adjacent solid (s) and fluid (f) cell, φ_i^ss the bond between two adjacent solid cells and φ_i^ff the bond between two adjacent fluid cells. The energy term φ_i^ff suggests that there are different bonds between solvent particles. In a well mixed solution this can, obviously, not be the case, however, for uniformity we will keep this notation. For crystals grown from a vapour phase this relation will reduce to Φ_i=$\text{1}\text{2}$φ_i^ss, because no fluid is present and thus φ_i^sf=φ_i^ff=0. However, for all other crystal growth systems, e.g. crystal growth from melt and solution, the bonds are defined by the overall bond Φ_i as in Eq. (6). The value of φ_i^ss can be calculated using force field calculations, but the values of φ_i^sf and φ_i^ff are not known. For this reason the proportionality condition [18]is introduced which states that the ratios between the different Φ bonds are equal to that of the different φ^ss bonds, which are known

$$\text{Φ}{}_{\mathrm{i}}\text{:}\text{Φ}{}_{\mathrm{j}}\text{:}\text{…=φ}{}^{\mathrm{<mtext>ss</mtext>}}{}_{\mathrm{i}}\text{:}\text{φ}{}^{\mathrm{<mtext>ss</mtext>}}{}_{\mathrm{j}}\text{:}\text{…}$$

According to this assumption, the ratios between the overall bonds Φ can be directly derived using the ratios between the solid–solid bonds φ^ss. We note that this is an ad hoc assumption without proof. Deviations from the proportionality condition can lead to more or less anisotropic lattices and a change in relative bond strength. However, it has been shown that for pseudo-hexagonal lattices, such as {1 1 0} and {1 1 1} faces of orthorhombic n-paraffin crystals, small deviations from this proportionality condition do not lead to large deviations in θ^R [24].

The enthalpy of dissolution ΔH_diss contains information about the values of the overall bonds Φ in the bulk of the crystal [36], because

$$\text{Δ}\text{H}{}_{\mathrm{<mtext>diss</mtext>}}\text{=}\text{∑}\text{i}\text{Φ}{}_{\mathrm{i}}\text{.}$$

Using the dissolution enthalpy, the solid–solid bonds φ^ss and

$$\text{Φ}{}_{\mathrm{i}}\text{=}\text{φ}{}^{\mathrm{<mtext>ss</mtext>}}{}_{\mathrm{i}}\text{2E}{}_{\mathrm{<mtext>cr</mtext>}}\text{Δ}\text{H}{}_{\mathrm{<mtext>diss</mtext>}}\text{,}$$

the overall bonds in the bulk of the crystal can be calculated. In Eq. (9)E_cr equals $\text{1}\text{2}$∑_iφ_i^ss and can be calculated using molecular mechanics. The experimentally determined roughening temperature T_{h k l}^R of a face {h k l} is determined by the values of the overall bonds Φ in the interface. If the critical dimensionless temperature has been calculated with solid–solid bonds which have been scaled to the strongest bond φ_str^ss, the strongest bond Φ_str can be calculated from the experimental roughening temperature using Eq. (2). Thus, the value of the strongest bond in the bulk of the crystal Φ_str^bulk can be calculated via ΔH_diss and Eq. (9)and the value of the strongest bond in the interface of face {h k l} Φ_str^int can be calculated via T_{h k l}^R and Eq. (2). This leads to three different cases which are usually discussed using a wetting parameter 37, 38:

$$\text{κ=}\text{Φ}{}^{\mathrm{<mtext>int</mtext>}}{}_{\mathrm{<mtext>str</mtext>}}\text{Φ}{}^{\mathrm{<mtext>bulk</mtext>}}{}_{\mathrm{<mtext>str</mtext>}}\text{=}\text{ln}\text{x}{}^{\mathrm{<mtext>int</mtext>}}\text{ln}\text{x}{}^{\mathrm{<mtext>bulk</mtext>}}\text{,}$$

where x^int is the concentration of the solute particles at the interface and x^bulk is the concentration of solute particles in the bulk of the mother phase. The case κ=1, Φ^int=Φ^bulk and x^int=x^bulk is considered as an artificial reference state and is called the equivalent wetting condition. If κ<1, Φ^int<Φ^bulk and x^int>x^bulk, wetting is considered more than equivalent and the bonds in the interface are weaker than in the bulk, due to the fact that φ^sf is more negative (or that φ^ff is less negative). The concentration of solute particles will be higher at the interface of the crystal than in the bulk of the mother phase. Finally, less than equivalent wetting is defined as κ>1 and thus Φ^int>Φ^bulk and x^int<x^bulk and the bonds in the interface are stronger than those in the bulk of the crystal.

In this paper the roughening transitions of the {1 1 0}/{1 1 1} faces of n-C₂₃H₄₈ and n-C₂₅H₅₂ crystals grown from mixtures of n-hexane and toluene will be studied as a function of the solvent composition. The focus will be on the actual roughening temperatures, the nature of the roughening transitions, the enthalpy of dissolution and the wetting of n-C₂₅H₅₂ and n-C₂₃H₄₈ in the dependence of different mixtures of solvents.