activated complex theory of barite scale control processes

Molecular Engineering7: 491–514, 1997. 491c 1997Kluwer Academic Publishers. Printed in the Netherlands.

Activated Complex Theory of Barite Scale ControlProcesses

MARIO BLANCO1,? YONGCHUN TANG2, PATRICK SHULER2 andWILLIAM A. GODDARD III 11Materials and Process Simulation Center, California Institute of Technology, Pasadena CA 91125,U.S.A.2Chevron Petroleum Technology Company, P.O. Box 446, La Habra, CA 90633, U.S.A.

(Received and accepted: 5 August 1997)

Abstract. A theoretical framework useful for the estimation of scale dissolution rate constants isintroduced. The model consists of (a) a dissolution mechanism, (b) a quantum mechanical force fieldcapable of describing the bulk and surface properties of barite, (c) a complete study of the structureof the dissolver molecule complex in solution, (d) a three dimensional periodic system useful formapping the barite-dissolver interactions, including the localization of the activated complex and (e) arate expression to estimate the dissolution rate constants from the properties of the activated complex.Our results show that molecular modeling, through a combination of molecular mechanics and highlevel quantum mechanical calculations, provide a new and insightful information about scale controlprocesses.

Key words: Barite, BaSO4, scale control, dissolution, inhibition, molecular modeling, activatedcomplex, scale in oil and gas production

1. Introduction

The deposition of insoluble mineral salts on the surface of water transport equip-ment is a widespread problem. Oil pipes, cooling towers, pumps as well as everydayplumbing are seriously affected by the accumulation of mineral deposits. Theseare mainly calcium carbonate (calcite) and barium sulfate (barite). These depositslead to equipment failure and costly repair and maintenance shutdowns. For exam-ple, water-born deposits cause significant problems in oil and gas production. Thevolume of saline water produced in oil and gas wells often is several times that ofthe hydrocarbons recovered. These deposits can restrict fluid flow in the reservoir,wellbore, or surface equipment.

Scale deposition from oil field ‘brines’ (typically 1–20% dissolved salt) can becaused by significant changes in the system temperature and pressure associatedwith hydrocarbon production. Another common mechanism for scale deposition isthe mixing of separate incompatible brines, one charged with multivalent cationsthe other rich in sulfate or carbonate anions. Consequently, the oil industry employspreventive (scale inhibitors) and maintenance (scale dissolver) treatments. Even the

? Corresponding author.

492 MARIO BLANCO ET AL.

better scale inhibitor programs do not completely prevent the unwanted accumula-tion of scale deposits in the well or surface equipment. Removal of barium sulfatescale is of particular concern because there are no known inexpensive chemicalreagents that can dissolve these deposits.

Two of the most common kinds of molecules used in scale treatments areaminophosphonates (inhibitors) and aminoacetates (dissolvers). The molecularmechanisms are believed to be (a) a strong surface binding process for scaleinhibition and (b) a strong cation chelation process for scale dissolution. We pointout here that these thermodynamic equilibrium processes are necessary but notsufficient to guarantee a successful treatment. For practical reasons the rate of dis-solution is more important and more difficult to estimate than the final equilibriumbetween the divalent cation, Ba+2, and the aminoacetate chelating agent (EDTA,DTPA).

Our purpose is to introduce a theoretical molecular framework useful for theestimation of scale dissolution rate constants. We cast our model borrowing ideasfrom activated complex theory which assumes that the passage through a transitionstate limits the rate of a reaction. Some experimental evidence, such as activationenergies, has been presented elsewhere [1]. Theoretical evidence in support of thepresence of dissolution energy barriers, which control the kinetics of the baritedissolution process, is presented for the first time here.

The molecular analysis begins with the derivation of an accurate force fieldfor barite, barium sulfate. The force field is capable of reproducing spectroscopic,mechanical, bulk and surface properties of barite. This is accomplished with arelative small number of molecular meaningful parameters (force constants, atomiccharges, van der Waals parameters) obtained from high level quantum mechanicalcalculations.

Quantum mechanical calculations are also presented to clarify the structure ofthe scale dissolver complex in solution. These were carried out to elucidate themost likely conformation of the dissolver molecule on its approach to the baritesurface. A single conformation, with significant lower energy than any other, wasidentified, indicating a much lower entropy of the approaching dissolver moleculethan previously expected. High level quantum mechanical calculations involvingcations are presented.

Our barite model of the 001 surface was sampled with the most stable EDTAcomplex. A map at 0.1 Å resolution uncovered previously unknown binding chan-nels for EDTA on the barite surface. The channels are formed by the intersection ofthe 001 and 100 planes, running parallel to the crystallographicb axis. The EDTAcomplex binds to these channels at a 56� angle to maximize its binding ability.

Molecular dynamics simulations in the presence of explicit water moleculeswere also conducted, but will be presented elsewhere. Instead, we use in this papera simplified model, based on molecular mechanics minimization. This simplifiedprocedure indicates no barrier to binding EDTA complex to the surface. It also

BARITE SCALE CONTROL PROCESSES 493

Figure 1. Four examples of Polyaminoacetate scale dissolvers. DTPA and EDTA are commer-cial products. DOTA and CDTA where uncovered during this investigation.

uncovers the presence of two energy barriers during dissolution, one related toBa2+ capture and the other to monovalent cation release.

2. Structural Characteristics of Scale Dissolvers

A few examples of chemical structures of polyaminoacetate scale dissolvers areshown in Figure 1. Diethylenetrinitrilopentaacetic acid (DTPA) is regarded asthe most effective scale dissolver by the oil industry. The dissolution kinetics ofDTPA have been extensively studied [1]. Ethylenediaminotetraacetic acid (EDTA)is perhaps the most studied chelating agent and has widespread use as scale dis-solver. 1,4,7,10-tetraazacyclododecane-N,N0,N00,N000-tetraacetic acid (DOTA) hasthe highest Ba2+ chelation constant of any known ligand. We discover this mole-cule as a potential scale dissolver while searching over four thousand compoundsin the NIST critical constants data base [2]. Since Ca2+ chelation constants aremore abundant than the corresponding Ba2+ for any given ligand, we developedthe following regression analysis equation between them:

logKML (Ba2+) = 0:7 � logKML (Ca2+) + 0:236 (1)

withR2 = 0:875. We found DOTA to be nearly indistinguishable from DTPA underall relevant conditions. However, DOTA is prohibitively expensive perhaps because


it is not made in large commercial amounts. Finally, cyclohexyldinitrilotetraaceticacid (CDTA) is a good example of how the geometry of EDTA can be constrainedto achieve a higher chelation constant, but not necessarily a higher dissolutionperformance.

The dissolution performance of these related compounds was measured by theconcentration, mg/l of Ba2+, dissolved in 10 g of 0.18 M dissolver test solutionfrom a 0.2 g sample of mineral grade barite (Potosi, Missouri) after three dayexposure at 60�C on a stagnant solution. The values are as follows [2]:

DOTA � DTPA (3500) > EDTA (3075) > CDTA (1120); (2)

while the high pH Ba2+ chelation constants, logKML , are ordered as follows:

DOTA (12.87) > DTPA (8.78) > CDTA (8.69) > EDTA (7.86): (3)

It is worth noting that DOTA has an equilibrium constant over ten thousand timeslarger than any other polyaminoacetate ligand towards Ba2+, including DTPA.Nonetheless DOTA’s dissolution performance is no better than that of the commer-cial dissolver DTPA. Note also that CDTA is nearly ten times a better chelant thanEDTA but a poor scale dissolver.

Clearly there is more to dissolution efficiencies than a strong chelation equilibri-um. Chelation constants are equilibrium constants, measured at low concentrationsunder mild ionic strength conditions. What is more important, these constants mea-sure the ability of the chelant to entrap Ba2+ cations which are free in solution, notfrom a strongly bound ionic solid. Consequently, logKML values are only of limit-ed use in identifying potential effective scale dissolvers. One may say that a strongchelation constant (logKML > 7) is a necessary but not a sufficient condition fora good scale dissolver.

3. Homogeneous and Heterogeneous Equilibrium Constants

All commercial scale dissolvers have multiple acid dissociation constants. Thedeprotonation of the tertiary amine (pKa > 7) is perhaps a crucial step in the finalchelation of the metal cation. For this reason most dissolution protocols call forhigh pH (�12). The high pH solution equilibrium is represented as follows:

L�m(aq) + M+q(aq) [ML ]q�m(aq) (4)

KML = [ML ]q�m(aq)=[L]�m(aq)[M]+q(aq): (5)

Typically q = 1, 2, 3 andm varies between 4 and 6. In high pH solutions the ligandis fully deprotonated, however it may still be combined with one or more cations.Since hydroxides of monovalent metal cations, sodium and potassium, are used toraise the pH, one must consider the equilibrium between these and the deprotonatedligand.


The binding equilibrium constants (KML ) are determined experimentally indilute solutions. On the other hand, barite dissolution involves more than twophases. Some elementary reactions take place at the boundary between the solidand the solution interface. Thus, dissolution is a heterogeneous reaction. We have toconsider the role of the interface in the mechanism in a different way than we haveconsidered the ‘homogeneous’ reaction, Equation (4). The molecular properties ofthe interface, at which heterogeneous reactions occur, are quite different from themolecular properties of the same substance away from the interface. The propertiesof the interface prove more difficult to be studied. The main obstacle is the sortingout of the elementary reaction steps that occur on the barite interface. Frequentlysome of the elementary reactions are diffusion controlled either to, from, or at theinterface. Experimental data, such as temperature dependent rate constants (see thefollowing section) indicate that desorption from the surface is the rate determiningstep in dissolution. Our molecular modeling results, presented in the followingsections, support these conclusions.

The heterogeneous equilibrium between the phases involved in the overall dis-solution process can be written as:

L(aq) + SKad�! LS(ad)

Kde�! LS0 (ad) (6)

where (aq) stands for aqueous environment and (ad) the surface adsorbed state.L is the ligand and S and S0 indicate a binding site on the surface and desorbedrespectively. Typically the binding site involves one or more metal cations and itshould formally be treated as a distinct species. For simplicity we may leave out theprime designation on the aqueous species. It should be kept in mind that the natureof the binding site can be quite different from the bound metal cation in solution.

According to the activated complex theory [4] the equilibrium constants forthese reactions can be written in terms of the partition functions per unit volume,qLS, qL andqS. Each of these partition functions is referenced to their individualzero of energy, and a Boltzmann factor which adjusts them to a common zero ofenergy, the so called zero point energy correction:

Kad = [qLS(ad)=qL(aq)qS]exp�[�Ead=RT]; (7a)

where the adsorption process,

�Ead = Ead[LS(ad)]� (Eaq[L(aq)] +Eaq[S]) (7b)

is typically exothermic. The equilibrium constant for the desorption process, typi-cally endothermic, is given by:

Kde = [qLS(aq)=qLS(ad)]exp�[�E=RT ]: (8)

Combining Equations (7a) and (8) we get:

Kde = [qLS(aq)=(KadqL(aq)qS)]exp�[(�Ede+�Ead)=RT ] (9)


Note that in this expression the desorption equilibrium constant does not dependexplicitly on the partition function of the adsorbed state. Thus, one might be temptedto replaceKad with the solution binding constant,KML , thereby equating S = S0.As we shall see, this is not always possible.

The binding geometries of the adsorbed ligand can be drastically different, andso will their partition functions, from the geometries in solution. This intermediatebinding state, S, is responsible for the lack of a linear relationship between disso-lution performance and solution equilibrium constants. The dissolution efficiencyof a given dissolver molecule is a strong function of the geometry of the adsorbedstate, as well as the geometry of the desorbing activated complex.

4. Activated Complex Theory of Dissolution Kinetics

The partition function of L, S, LS and LS0 can be obtained using the ideal gas,rigid rotor, harmonic oscillator approximations to the molecular energy levels,with vibrational force constants and geometries preferably determined fromabinitio quantum mechanical calculations. From the knowledge of the reaction pathcoordinate one may also obtain all the necessary information to calculate thepartition function of the activated complex. Factoring into a product of translational,electronic, rotational and vibrational partition functions, we have:

qyLS = q

yt q

yeq

yr q

yv: (10)

The first three partition functions present no special computational problems. Thevibrational partition function can be written as:

qyv =3N�7Y

i=1

1=(1� exp[��y�;i=T ]); (11)

where

�y�;i = h�

yi =k: (12)

The set of frequenciesf�yi g for the activated complex can be calculated by analo-gy with the vibrational frequencies of stable molecules. First one must determinethe saddle point between reactants and products, where a single imaginary vibra-tional frequency is found. Semi-automatic procedures for this operation are widelyavailable at both semi-empirical andab initio levels of calculation. Care must beexercised with the low frequency vibrations, since these contribute significantly tothe partition function of the activated complex. The activated complex has weakvibrations as well, in particular the bending vibrations perpendicular to the reac-tion coordinate, and thus parallel to the barite surface. Estimates of these surface


parallel frequencies could be the main source of error in the calculations ofqyLS.

The reaction rate constant, in cm3/molecules, can be written as:

k(T ) = �(kT=h)(qyLS=qad)exp(��Ey

Z=kT ); (13)

where� is a transmission coefficient, normally around unity, and�EyZ is the zero

point corrected activation energy.

�EyZ = Eb + (E

yZ �Ezad): (14)

In Equation (14)EyZ andEzad are the zero point energies of the activated complex

and the adsorbed state, respectively. The diagram in Figure 2a illustrates thesequantities.

Equation (13) formally represents the dissolution rate constant. Note thatk(T )is completely independent of the solvated state LS(aq). Again, this indicates theimportance of surface adsorption and desorption processes as compared to thethermodynamic equilibrium in solution. A simple model can be derived with somesimplifying assumptions:

(a) adsorbed molecule and activated complex undergo no rotations;(b) harmonic oscillator approximation for all vibrations on the non-degenerate

electronic ground state;(c) adsorbed molecules undergo two low frequency vibrations parallel to the

surface (�xy and�yx).(d) all other internal vibrations have the same frequencies in the adsorbed state as

in the activated complex. The vibration perpendicular to the surface becomesa translation along the reaction path.

With these assumptions we have that:

qyLS � q

yt q

yv (15a)

= (2�mkT=h2)1=2=f(1� exp[�h�yxy=kT ])(1� exp[�h�yyx=kT ])g (15b)

and when combined with Equation (13):

k(T ) = �(kT=h)(2�mkT=h2)1=2(T )exp(��Eyz=kT ); (16)

where

(T ) =(1� exp[�h�o

xy=kT ])(1� exp[�h�oyx=kT ])(1� exp[�h�o

z=kT ])

f(1� exp[�h�yxy=kT ])(1� exp[�h�yyx=kT ])g

(17)


Figure 2. (a) Potential energy diagram illustrating the zero point energies and the energybarrier during barium cation dissolution. LSad is the adsorbed ligand species on the baritesurface while LSaq is the ligand plus cation in solution. (b) The geometry of the dissolvermolecule and corresponding cation over the barite 001 surface. The diagram indicates thevibrational frequencies of the adsorbed state.


Neglecting terms on the order of exp(�2h�=kT ) we get:

(T ) �1� exp[�h�o

xy=kT ]� exp[�h�oyx=kT ])� exp[�h�o

z=kT ]

1� exp[�h�yxy=kT ]� exp[�h�yyx=kT ]: (18)

Equation (16) might be used to screen a potentially large number of scale dissolvers.One needs to compute the zero point correction,��Ey

z, and the five frequenciesof interest, the three frequencies�o

xy, �oyx and�o

z of the adsorbed state and thetwo perpendicular frequencies�yxy and�yyx of the activated complex. Figure 2billustrates the relevant frequencies.

5. A Hessian Biased (HB) Force Field Model for Barite

Before we proceed with theab initio modeling of these sorption and desorptionprocesses, a force field model for the barite surface greatly facilitates the iden-tification and characterization of the bound and activated complexes. The forcefield should also describe the dissolver molecule and the surrounding solvent andcounter ions. A force field can provide the zero point correction and necessaryfrequency information. However,ab initio quantum calculations are more accurateand become affordable once the initial (adsorbed) and intermediate (activated com-plex) states are well characterized at the force field level because fewer convergencesteps are needed.

The bulk and surface properties of barite can be calculated using a properlyparametrized force field. The force field gives the energetics of the crystal as afunction of changes in atomic positions. One may use experimental informationto parametrize a force field, quantum mechanicalab initio information or a com-bination of the two. An example of an empirical force field for the barite crystalhas been provided by Rohet al. [5]. We refer to this as Model 2. We employed theBiased Hessian approach of Dasgupta and Goddard [6] to develop a new force fieldfor barite. We include the unit cell dimensions (a, b andc) of the crystal as the onlyempirical pieces of information to obtain this new force field. The unit cell angles,�, � and were allowed to vary but remained within 1� of the experimental 90�

values (Figure 3).Surprisingly this‘ab initio’ force field reproduces the structural crystal infor-

mation, bulk density, the nine known elastic stiffness constants, the quantum vibra-tional frequencies of the sulfate anion (Figure 4) and surface energies consistentwith the morphology of barite (Figure 5). In addition our model (Model 1) solves the‘collapse’ problem present in molecular dynamics runs with Model 2. Moleculardynamics simulations of bulk barite with Model 1 are quite stable. The analyticalforms of the potential and the parameters for Model 1, which were used in all thecalculations reported here, are included in Table I.

The empirical force field Model 2 also gives good estimates for the elastic con-stants and crystal structural parameters for barite. The parameters in the potential


Figure 3. Barite unit cell parameters predicted with Model 1 (this work) and Model 2 (Ref.4). Unit cell contains 4 Ba cations and 4 corresponding Sulfate anions.


Figure 4. Ab initio calculated vibrational spectrum of the sulfate anion. Predicted frequenciesare calculated using Force Field Model 1 and 2.

and the charges were refined by fitting the structural and the elastic stiffness con-stant of barite, using a least square procedure. Intermolecular potentials (Ba—Oand O—O) were originally represented with the Buckingham potential:

V (r) = Aexp(�r=�)� C=r6: (19)


Figure 5. Predicted surface energetics for various low Miller index surfaces of barite for Model1 and Model 2.

Unfortunately this form leads to parameters with no physical meaning. For example,the pre-exponential factorA has a value of 103585.02 eV in the O—O potentials


Table I. Barite force field Model 1

ChargesBa (2.0) formal chargeS (1.544)O(�0.886) from LMP2 LAV3P** PSGVB

Valence termsS—O Bond Stretch: MorseEb = D0[e

��(R�R0) � 1]2 with � = (Kb=2D0)

Kb = 932:3884 kcal/mol-ÅR0 = 1:5050 ÅD0 = 75:3987 kcal/molO—S—O Angle bend: Cosine–stretchEab = 1=2K�(cos� � cos�0)

2= sin2 �0

�K�R[(R1 �R0) + (R2 �R0)](cos� � cos�0)= sin� +KRR(R1 �R0)(R2 �R0)

K� = 326:1375 (kcal/mol)/radian2 K�R = 32:1793 (kcal/mol)/(Å-radian)KRR = 19:0062 (kcal/mol)/Å2 �0 = 109:4712 degreesCross term angle-angle inversionEaa = 1=2K��(cos�1 � cos�0)(cos�2 � cos�0)

K�� = 86.9367 (kcal/mol)/(radian2)�1 corresponds to theijk angle,�2 corresponds to thejkl angle.Non-bond terms

R0 D0 �

O—O exponential-6 (a) 3.310 0.310 16.70Ba—O pure exponential (b) 3.542 0.572 11.51(a)Enb = D0f[(6=(� � 6))e��(1�R=R0)]� [(�=(� � 6))(R0=R)

6]g

(b)Enb = D0e�(1�R=R0)

of reference [5]. We have reparametrized these potential using the exponential-6and the pure exponential forms:

V (R) = D0f[(6=� � 6)exp(�(1�R=R0)]� [(�=(� � 6))(R0=R)6]g (20)

V (R) = D0 exp(�(1�R=R0) (21)

for O—O and Ba—O non-bond potentials, respectively. These parametrizationslead to values that can be interpreted in terms of binding energies (D0) and equi-librium distances (R0). Table II contains the reparametrized potential.

Table III compares the experimental values of the barite elastic stiffness con-stantsCij with the predictions from Model 1 force field (present work) and the fittedvalues with force field Model 2 [5]. Model 1 predicts the barite elastic stiffnessconstants correctly without the need to use this information to obtain the force fieldparameters. Onlyab initio quantum mechanical information has gone into fittingthe force field in Model 1.


Table II. Barite force field Model 2. This is a reparametrization of ModelII in Ref. [4]

Charges:Ba (2.0) S(1.36) O(�0.84)

Valence termsS—O Bond stretch: MorseEb = D0[e

��(R�R0) � 1]2 with � = (Kb=2D0)

Kb = 332.1936 kcal/mol-ÅR0 = 1.5050 ÅD0 = 115.3450 kcal/molO—S—O Angle BendEab = 1=2K�(� � �0)

2

K� = 346.035 (kcal/mol)/radian2

�0 = 109.47 degreesNon-bond terms

R0 D0 �

O—O exponential-6 3.28816 0.30114 16.44079Ba—O pure exponential 3.4884 0.59869 12.00Enb = D0f[(6=(� � 6))e��(1�R=R0)]� [(�=(� � 6))(R0=R)

6]g

Enb = D0e�(1�R=R0)

Table III. Barite elastic stiffness constants,Cij ingigaPascals.

ij Experiment FF Model 1 FF Model 2(this work) (Ref. [4])

11 95.1 96.4 101.312 51.3 43.5 44.213 33.6 35.4 27.822 83.6 92.7 87.423 32.8 32.3 29.633 110.06 115.2 117.544 18.1 17.7 16.455 29.0 35.8 35.266 27.7 27.1 28.4

rms: 4.56 GPa 5.30 GPa

6. Structural Characteristics of EDTA in Solution

KOH and NaOH are generally used to raise the pH of a solution above pH =12. Although the aminoacetate ligand is fully deprotonated there is a sufficientnumber of monovalent cations (K+, Na+) with which the ligand can be associated.It is the monovalent coordinated species that approaches the surface of barite in thedissolution process. Consequently it is very important to understand the equilibriumof these complexes.


We screened a large number of local minima for EDTA complex with fourmonovalent cations (Na+ and K+). The multiple local minima were obtainedthrough the use of quenched dynamics and the Universal Force Field [7]. We usedformal charges on the monovalent cations and the charge equilibration procedure(Qeq) of Rappe and Goddard [8] to indicate atomic charges on the ligand.Qeq

charges are sensitive to the geometry and the proximity of the cations. Therefore,we used a self-consistent procedure to re-calculate the atomic charges when a newminimized structure was obtained, until the total RMS force on the atoms was lessthan 0.12 kcal/mol Å.

Results for sodium and potassium are shown in Table IV. Each of the structurescan be characterized as follows. Each of the monovalent cations is identified bythe list of atomic close contacts it makes with the ligand molecule. A close contactis defined between atomi andj if these are closer than the sum 0.89* (Ri + Rj),whereRi andRj are the van der Waals radii of the two atoms. The factor 0.89scales the radii to the point where a Lennard–Jones potential becomes positive (therepulsive wall of the potential).

The stability order of the complexes is quite similar, even though the cationshave widely different ionic radii, 1.38 Å for K+ and 1.02 Å for Na+ [9]:

K+ : A < D < B < E < F < G < C (22a)

Na+ : A < B < D < E < F < G < C: (22b)

The four LM4 most stable complexes are depicted in Figure 6 for K+. In bothcases the lowest complex is defined by structure A. In both cases the secondcomplex is much higher in energy to be significantly populated at room temperature.Consequently one would conclude that the LM4 complexes of EDTA have a welldefined and stable geometry in solution. This is quite an unexpected result.

The most likely conformation of EDTA on its approach to the barite surface isstructure A for any monovalent cation. Thus, the approaching dissolver moleculehas a low conformational entropy in solution. A well defined solution complexgeometry could be interpreted as a pre-forming state of the ligand before thisbinds to the barite surface. As such, it offers some new design ideas for futurescale dissolvers, as well as new insights into the nature of the chelating agents insolution.

7. The Binding Process

The dissolver molecule must reach the barite surface to react. A model of thedissolution process must, by necessity, include a surface binding step. A finitesurface model, unless carefully constrained cannot duplicate the required physics.It is necessary to use periodic boundary conditions to include all of these featuresand to avoid ‘edge’ artifacts of finite systems.


Table IV. UFF/Qeq stability of EDTA: M4 complexes.

Conf. DE Na1 Na2 Na3 Na4 Energy Coulomb vdW Internal

vdw close contactsa

A 0.0 24N1 134 57N2 568 �822.5 �907.5 55.5 29.5B 58.8 24 13 57N2 568 �763.7 �849.0 55.0 30.3C 139.7 25N1N2 134 68 Pb 641.9 �682.8 29.9 11.0D 63.2 25N1 134 68 NPb �759.3 �795.9 23.6 13.0E 64.3 24N1N2 13 56 678 �758.2 �810.9 39.3 13.4F 69.3 247N1N2 13 56 68 �753.2 �801.1 36.4 11.5O 133.0 247N1N2 13 257N2 68 �689.5 �745.4 41.9 14.0

Conf. DE K1 K2 K3 K4 Energy Coulomb vdW Internal

A 0.0 24N1 134 57N2 568 �748.5 �831.1 54.5 28.1B 56.7 24 13 57N2 568 �691.8 �771.3 53.0 26.5C 160.4 25N1N2 134 68 P �588.1 �634.2 34.8 11.3D 39.4 25N1 134 68 NP �709.1 �751.9 31.4 11.4E 66.4 24N1N2 13 56 678 �682.1 �733.2 38.1 13.0F 71.8 247N1N2 13 56 68 �676.7 �732.8 41.5 14.6G 145.7 247N1N2 13 257N2 68 �602.8 �653.1 37.3 13.0

a Van der Waals close contacts at scale radius 0.89.b P and NP indicate the metal cation is located on the polar and non-polar side of the ligand respec-tively.

Periodic boundary conditions were imposed on a 2� 4 � 2 unit cell. Thec axis was then extended to 30 Å, exposing the 001 surface. This gives enoughroom for one dissolver molecule and up to 175 water molecules. The system wasallowed to equilibrate for 50 ps at 60�C under NVT conditions. We kept thebottom portion of the barite (two sulfate lower layers) fixed, but allow the toptwo layers of barite, the dissolver molecule, solvent molecules and counter ions tomove freely. The dissolver molecule (EDTA) found a resting position on the 001


Figure 6. Four distinct conformations of EDTA and corresponding K+ monovalent coun-terions. Conformation A is significantly lower in energy than the others. This is a stretchedconformation which facilitates the binding of the Ba+2 divalent cation upon binding to thebarite surface.

plane. Finally the unit cell was minimized to an rms force less than 0.1 kcal/mol Å.Upon minimization the EDTA and its four counter ions maintained the geometrydesignated A in Table IV.

With the molecule in its bound position a systematic 2D grid sampling wasconducted by displacing the center of mass of the EDTA molecule over the 001surface in increments of 0.1 Å. The EDTA molecule was allowed to ‘hover’ overthe surface such as to avoid any van der Waals overlaps. This was accomplished bysolving all excluded volume constraints for each of the translations on the surfaceaccording to the Molecular Silverware method [10]. The total energy at each of the


Table V. Variation of the energy of the periodic system with the distance,along the site of highest binding of the EDTA molecule to the 001 surfaceof barite.

Distance (Å) to 001 surface Energy (kcal/mol) of periodic cell 001

10 �27,546.1020 �27,563.6130 �27,563.3840 �27,563.44

grid points was recorded after subtracting a constant value from all the energies,the zero energy level.

The zero of energy is defined by lifting the molecule at the site of highest bindingaway from the barite surface a given distance. The process is repeated until theenergy of the periodic system converges to a fixed value. The values for 10 to 40 Åare shown in Table V. Thus, by 20 Å the energy has converged within 0.3 kcal/mol.This value was then subtracted from the grid energy values to give effectively arelative binding energy.

A contour map of the binding energies on the 2D grid is shown in Figure 7a.Figure 7b shows the EDTA molecule in its strongest binding geometry. Note thatthe molecule binds to the surface at an angle of 56o with respect to theb axis. Tworows of Ba+2 ions are aligned along the b crystallographic direction, beneath theEDTA surface bound molecule. The contour map shows deep channels runningparallel to the location of the Ba+2 ions rows along theb axis. The channels areapproximately 23 kcal/mol deep, the binding energy of the EDTA molecule in theshown geometry. Perpendicular to the channel (010 direction) large energy barriers(15 kcal/mol) prevent side motion of the EDTA molecule. In some places a steepenergy walls raise up over 28 kcal/mol in little over 3 Å displacement of the centerof mass of the molecule. Only small “bumps” (3 kcal/mol) separate the periodicrow of the global minima along theb axis. At or near the top of the “hills” thesurface becomes repulsive (4 kcal/mol) when compared with the EDTA molecule40 Å away from the barite surface. At these locations the EDTA molecule wouldactually be pushed upwards, away from the surface.

We conclude that the binding of EDTA on the barite 001 surface is highlyspecific, the binding sites being located along the well defined channels runningparallel to theb axis. Movements along directions other than theb axis are highlyrestricted. The EDTA molecule maximizes the interaction with the 001 surface byplacing its molecular axis at 56� angle with respect to theb axis. In this positionthe acetate moieties of the amino groups are capable of binding to two partiallyexposed Ba+2 ions.


Figure 7. (a) Potential energy contour plot of the binding of EDTA to the 001 barite surface.Notice the presence of a global minimum, with a binding energy of�23.4 kcal/mol. A highenergy barrier severely limits motion of the dissolver molecule along thea axis. A shallowbarrier, 3 kcal/mol, exists along theb axis. (b) Corresponding molecular display of EDTA on001 barite surface. The molecule is shown in the global minimum. Notice the 56� angle thatthe axis of the molecule makes with respect to the crystallographicb axis. This enable EDTAto bind to two rows of barium cations instead of only one.


Figure 8. EDTA: Ba dissolution profile showing the energetics and main events in a dissolutionprocess. The first barrier corresponds to the capture of the barium cation. A second barrier wasfound that corresponds to the monovalent cation release to the 001 surface.

8. The Dissolution Process

Experimental data on DTPA indicates the existence of a dissolution activationenergy significantly larger (13.5 kcal/mol) [1] than expected for a purely diffusioncontrolled heterogeneous reaction (<4.0 kcal/mol). Theoretical evidence for theexistence of such a barrier is presented here for the first time. The periodic modeldescribed in the previous section provide the basis for probing the nature of themolecular interactions during the elementary steps leading to dissolution.

The barite model, four barium sulfate layers thick, was allowed to equilibrate(two top layers). A series of displacements, each of 0.1 Å long, perpendicular to the001 surface were applied to the center of mass of the EDTA complex. The originalgeometry correspond to the binding geometry shown in Figure 7b. At each step thecomplex was allowed to relax after the relaxation of the barite was completed. Thetotal energy was recorded at each step and the lowest point on this pathway definedas the zero of energy. The results are shown in Figure 8.

After a total displacement of about 12 Å the complex had captured one Ba+2 ionand released two monovalent ions. The activated complex geometry is shown inFigure 9, together with the previous positions of the Ba+2 ion before capture. Theenergy barrier for Ba+2 capture was calculated to be 18.8 kcal/mol. This calculated


Figure 9. A superimposed figure of the process of barium abstraction by EDTA from the 001surface of barite. Positions 1–4 are shown as well as the positions of the monovalent cations atthe end of the dissolution process.

value for EDTA is larger than the experimental effective dissolution activationenergy of DTPA.

A second and much larger barrier was found in the reaction pathway. This barrier(55 kcal/mol) corresponds to the temporary loss of favorable Coulomb interactionsduring the release of two monovalent ions by the EDTA complex after the Ba+2

capture event. We expect this barrier to be significantly lowered once the EDTAcomplex and the barite surface are modeled with a dynamic water environmentaround them. This is the subject of ongoing work.

9. Future Work

There is some evidence to indicate that the dissolution process may take place atcrystal edges and defects sites. Evidence is found in the kinetic studies of Putniset al. [1]. The dissolution rate was found to be directly proportional to the squareroot of the total surface area exposed, suggestive of a linear relationship betweendissolution rates and the amount of crystal plane edges in the sample. Strongerevidence comes from dissolution pits identified using AFM microscopy [11].


Figure 10a. A water dissolution pit geometry on the 001 barite surface is bounded by the�210 and 210 planes.

Although the solubility product of barium sulfate is onlyKsp = 10�99 at25 �C [12], water still manages to form a small number of pits on the 001 surface.These pits are bounded along the 210 and�210 directions (Figure 10a). Theseexperimental studies have concluded that the DTPA dissolver molecule takes overwhere water dissolution has begun, but it quickly alters the geometry of the pit.The pits have well-defined trapezoidal geometries (Figure 10b), with their paralleledges along the 100 direction, and the converging edges along the 110 and 1–10directions. Because the pits grow along theb axis, the 100 direction on the 001surface, it would appear that DTPA binds to the 001 barite surface in a similarmanner as EDTA.

In the edges of a pit the binding geometry of the dissolver molecule may changesomewhat. We speculate that the molecular axis may still be oriented along the 100direction but two of the aminoacetate groups may turn towards the interior of thepit.

It is also observed that pit growth decreases with time from an initial 6 nm/minto less than 2 nm/min within one hour. Future work will try to elucidate the originof this reduction in pit growth rates. One possibility is that the DTPA moleculesadsorbed on the pit edges eventually encounter other DTPA molecules which havebecome bound directly to the 001 surface. This over crowding of the surface haltsthe dissolution of the next row of Ba+2 cations and prevents the pit edge from


Figure 10b. The water dissolution pit in Figure 10 is modified by the action of a commercialscale dissolver, DTPA, creating new pits bounded by 100 and 110 Miller planes.

continuing to grow. These mechanisms of pit growth will be the subject of futuremolecular dynamics andab initio studies.

10. Conclusions

We have introduced a theoretical framework useful for the estimation of scaledissolution rate constants. Our model, cast within the concepts of activated complextheory, is fully atomistic. The model consists of (a) a dissolution mechanism, (b)a quantum mechanically derived force field capable of describing the bulk andsurface properties of barite, (c) a complete study of the structure of the dissolvermolecule complex in solution, (d) a three dimensional periodic system useful inmapping the barite-dissolver interactions, including the localization of the activatedcomplex and (e) a rate expression to estimate rate constants from the properties ofthe activated complex.

Experimental and now theoretical evidence supports the existence of activationbarriers, which are ultimately responsible for controlling the kinetics of the baritedissolution process.

The barite force field consists of energy expressions with physically meaningfulparameters obtained from high level quantum mechanical calculations.


High quality quantum mechanical calculations involving monovalent cationsconfirmed the structure of the scale dissolver complex in solution. The resultssuggest the existence of a pre-formed dissolver complex in solution, with a muchlower entropy than previously expected.

A binding map, at 0.1 Å resolution, of the barite-dissolver interactions uncoveredpreviously unknown binding channels for EDTA on the barite 001 surface. Thechannels are formed by the intersection of 001 and 100 planes, running parallel tothe crystallographicb axis. The EDTA complex binds to these channels at a 56�

angle to maximize its binding ability. There is no energy barrier for binding to thesurface. The geometry of the binding state on the 001 surface differs significantlyfrom the geometry of the EDTA : Ba+2 complex in solution. AFM analysis mightbe able to confirm these findings.

A simplified model, based on molecular mechanics minimizations, has con-firmed the existence of an activated complex. The calculated energy barrier forBa+2 capture by EDTA is slightly higher than the experimental effective activationenergy for DTPA.

We conclude that molecular modeling can, when combined with high levelquantum mechanical calculations, provide new and insightful information aboutscale control processes. These include (1) the bulk and surface properties of barite,(2) the geometry of the complexes involved in scale control, (3) the existence ofactivation energy barriers and (4) a detailed molecular information about the role ofthe counter ions in the dissolution process. We expect that this new model will beemployed by scale dissolvers manufacturers, or by heavy commercial users of thesechemicals. Use of the model should lead to new compounds for scale dissolutionand eventually to uncover new chemistries that may lead to faster, more efficientand economic compounds for all scale control processes.

References

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Department of Commerce (October 1993).3. P. Shuler: Chevron Petroleum Technology Company (private communication).4. K. J. Laidler and J. C. Polanyi: inProgress in Reaction Kinetics, Pergamon Press, Oxford, Vol.

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activated complex theory of barite scale control processes

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