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Mechanics of dispersions. Part 1.—Identification of parameters in structural hysteresis

 

作者: Allan J. B. Spaull,  

 

期刊: Journal of the Chemical Society, Faraday Transactions 1: Physical Chemistry in Condensed Phases  (RSC Available online 1977)
卷期: Volume 73, issue 1  

页码: 128-134

 

ISSN:0300-9599

 

年代: 1977

 

DOI:10.1039/F19777300128

 

出版商: RSC

 

数据来源: RSC

 

摘要:

Mechanics of DispersionsPart 1 .-Identification of Parameters in Structural HysteresisBY ALLAN J. B. SPAULLSchool of Chemistry, Brunel University? Uxbridge: MiddlesexRcceived 9th March, 1976A necessary and sufficient set of parameters that describe the mechanical properties of a perfectthixotropic dispersion is derived using a statistical method.By reference to a study of the viscoelastic and thixotropic properties of carbonblack in mineral oil, Mewis et a2.l have reported on the phenomenon of structuralhysteresis. This work is of interest because of the experimental proof that char-acterization of the structure of thixotropic systems by one parameter is inadequate, asubject about which there has been some controversy.2There are many reported examples of two phase systems of monodisperse and ofheterodisperse dispersions of particulate solids in a liquid phase showing visco-elasticity? or thixotropy, or both.3 The size of the particles has ranged from the lowerlimit to just above the upper limit of what is called the colloidal subdivision ofWhether the elastic response in such viscoelastic systems is entropic orpotential in origin has not been established.It is, therefore, of some importance toexamine the basic theory of two phase dispersions so as to determine the number ofparameters required to describe behaviour observed in the usual kind of rheologicalexperiment, and to inveytigate how the theory could further our understanding of suchdispersions and of related scientific problems. We choose a statistical thermodynamicmethod.ANALYSISThe analysis is based on one given by G~ggenheim.~ The preliminary step insolving a physicochemical problem is the choice of parameters required for an,adequate macrodescription of the system, preference being given to those whichaccord physical significance to underlying theory, and are easily determined experi-mentally.Although thixotropy has been defined somewhat differently by differentauthors,6* we will regzrd it as a time dependent reversible decrease of viscosity undershear, resulting from a reversible breakdown of structure.*-1° To this end, wedescribe a perfect thixotropic model which can be used to compare with real systems.Consider two discrete particles of colloidal dimensions in a liquid continuum. Theirsize is bounded and their (potential, distance) interaction is composite, arising fromattraction (London-van der Waals forces) and repulsion (electrical or steric forces orboth).ll The interaction energy function has two minima, primary and secondary,separated by a potential barrier.The interparticle distance depends on temperature,for if the depth of the secondary minimum is greater than several units kT of thermalenergy, the interaction leads to a permanent physical bond with a fixed interparticledistance between the two particles. On raising the temperature the interparticle12A. J. B. SPAULL 129distance increases until a point is reached when the interaction is so weak in relationto kT that the bond is considered as broken, the translational motion of the particlesbecoming independent of each other.Formation and breaking of bonds can bereversed by varying the temperature. Similarly, interparticle distance can be variedby applying i? shear field. Spzcifically, the physical bond distance between twoparticles in secondary interaction can be increased by increasing the shear field untilultimately the applied couple is sufficient to overcoine the interaction so as to break thephysical bond. To continue the development of the model, let there be N suchparticles in volume V. No restriction is placed on their size or shape, except thatthcj are c~lloidal.~ The barrier between the primary and secondary minima issufficient to prevent primary interaction between particles.Pairwise interactionQCCU~S between nearest neighbours, surrounding particles having no influence on theifiteraction between two given particles. The shear field has 110 effect on the solid-liquid surface structure of the particles, so electrical and steric repulsion potentialsare unchanged in a varying field.Depending on their thermal energy, someparticles will remain unbound, whereas others will form groups of two or moreparticles. Unbound particles will translate in a manner determined by the hydro-dynamic and Brownian interactions with the liquid continuurn, i.e., they could alsobe rotating. Groups of particles may also translate and rotate as a result of hydro-dynamic and Brownian interaction. Further, the geometric configuration of thegroups may be affected by hydrodynamic interaction with the continuum phase ; twolimiting shapes are spherical and linear.Particles located in a group may also berotating : indeed part of a group of particles may rotate in a different way from thatin another part of the same group. However, cz complete description of all possibleand actual physical processes of the particles is not necessary.The relation between the above microdescription and the macrodescription, forprescribed values of the inacroscopic parameters, N, V, energy (thermal plus shearfield) U, and a parameter c7 can be derived statistically by averaging over all accessiblephase space states. The properties of 5 are explored later. The dispersion isassumed to be an assembly of discrete colloidal particles in a liquid continuum,independently distributed in phase space, whenA shear field is applied at constant T.where Ur is the energy (thermal plus shear field) of phase space r, 5, the thermo-dynamic probability of the sth value of 5, Q(t,) a group of values of 5, of weight Rwhich we do not distinguish, and k the Boltzinann constant.Using a concentric cylinder geometry of infinite length, the dispersion is strainedat a constant rate.After infinite time, t (see appendix), < reaches a false thermo-dynamic equilibrium vaIue c:, when the microscale processes of the assembly nolonger change. The equilibrium energy, U", is given by :We conclude that, for i! given geometry, there is a one-to-one correspondence betweenthe equilibrium energies of the assembly and the constant rates of strain applied forinfinite time.1-I30 MECHANICS OF DISPERSIONSAt time to (= l), let us change the equilibrium energy of the assembly by changingthe rate of strain to a higher constant value, so that after infinite time the value ofU ; becomes Uh, and 5: becomes 5;.The instantaneous values of the energy and ofthe parameter 5 at time ti are respectively U: and 5'. At this higher rate of strain,certain changes in the microscale processes take place, compared with those previouslydescribed at equilibrium <:. For example, some bonds will no longer be stable andhence will rupture ; the interparticle distances between the particles forming asecondary bond may increase as a result of the higher potential energy they possess ;accordingly the geometric configuration of many groups will change.Thus, as wellas a change in the energy of the assembly, there is a change in the distribution of theenergy ; since there is a change in the thermodynamic probability of the phase spacesavailable, various values of 5, will contribute to the sum 5,. The change in dis-tribution of energy is time-dependent : physically it is a relaxation process. Oneimportant result of the relaxation is that even when the total energy of the assemblyhas reached a constant value, changes may still occur in its distribution among thephase spaces as changes in their thermodynamic probability occur, Le., we concludethat Ui does not necessarily take the same time to reach U i as ti does to reach <;.The parameter is a measure of the extent to which the assembly has approached athermodynamic equilibrium state.It is, therefore, appropriate to retain the nameDe Donder l 2 gave to the parameter 5, viz. the extent of reaction of a (thixotropic)physicochemical change.From our argument on the microdescription, four macroscopic properties areadequate to describe the rheology of a thixotropic dispersion; a suitable choice isvolume, composition, energy and extent of reaction.If 5 isknown, the parameter U can be determined from measurements of stress, p i j , sinceU = p,VN<, or in differential form (aU/aV),, = p i j , where p i j is a physicalcomponent of the stress tensor. There is difficulty in determining 5.In physico-chemical studies the thermodynamic probability, or partition function (to which < isrelated) is determined from spectroscopic measurement ; it may also be calculated.The analogue in mechanics is to obtain the mechanical spectrum by subjecting thesystem to a series of forcing vibrations having a wide range of frequencies, from whichthe relaxation spectrum may be determined. However, there is at present no simplerelation between 5 and the relaxation spectrum. Furthermore, there are almostcertain to be experimental difficulties : the mechanical technique is necessarilydestructive of what it is being used to determine ; a sufficiently sensitive experimentaldevice for determining a particular microscale process may be elusive.Values of ti, determined from transient spectra taken during a continuous thixo-tropic path from t = 1 to t = co, are required.On such a path the values lie betweenlimits, 0 < ti < I. A physical property of 5' is that it is the instantaneous descriptionof the microscale phase spaces, for prescribed values of N, V, and U ; the relationbetween ti, other macroscopic parameters, and the microscale processes can beobtained statistically and the relation is formally similar to eqn (1). Although wehave indicated that ti can be obtained from the spectral density function formechanical loss, it can, in principle, be obtained from any transient spectra resultingfrom perturbation of the microscale processes. Such perturbation can be achievedby a number of experimental techniques, and, depending on the method used, willgive different information about the physical nature of the microscale processes. Fora simple thixotropic change, e.g., from chains of particles bonded together to free,unbound, discrete particles, to be followed through 5, the spectral technique must beprecise and discriminating enough so as to follow the change in the statistical weightSThe next problem concerns the experimental determination of U and 5A.J. B. SPAULL 132of at least one of the microscale processes, such as the appearance of unboundparticles, or the disappearance of chains. An adequately accurate technique coveringa wide range of frequency would allow the determination of the time dependence of allmicroscale processes, but this is not needed in an evaluation of 5.One example of the mechanical spectra l3 is shown in fig.1. Transient mechanicalspectra taken with the dispersion at rest at one minute and 17 h intervals after thecessation of steady state shear (0.071, 0.71, and 7.1 s-I) are shown. The symbol His the relaxation spectrum (shear),14 and was determined from experiments using aWeissenberg rheogoniometer. In practical applications the time dependence of < isobtained from the variation in the character of spectra with time.log T / SFIG. 1.-Transient relaxation spectra taken during thixotropic build-up at one minute and 17 hintervals after the cessation of flow (0.071, 0.71 and 7.1 s-l), using a 2.7 % carbon black dispersionOther time dependent transient relaxation spectra have been reporfed,15 thus theapproach to dielectric spectra developed by Helsen et aZ.,16 may prove speciallysuitable in the determination of (.in mineral oil (w/w).After one minute -; after 17 h ---.DISCUSSIONWe apply our analysis to the assembly that we have described ; it closely resemblesthixotropic systems encountered in technology. The equilibrium curve, giving thevariation of U, as a function of t,, is shown in fig. 2. For all values of the thermo-dynamic equilibrium parameters dc/dt = 0 ; the function U, = UJ&) lies in thet = co plane, see fig. 2 and represents an infinite number of false thermodynamicequilibrium states. Most research work has been argued on the basis of equilibriumcondition^,'^ so it is not surprising that the one-parameter theory of thixotropy-adistinct one-to-one correspondence between macroscopic properties-has obscuredthe underlying fundamental problem.Non-equilibrium values of U, U', and t, ti,determined during thixotropic build-up and break-down at different values of t areshown in fig. 2 by projecting them on the t = 00 plane. During build-up and break-down, the one-to-one correspondence disappears, when correct analysis allows theemergence of the two-parameter thixotropic theory.Some paths that can be taken during break-down and during build-up are shown.The paths would depend on the mode of application of strain during these processes.For example, starting from the same equilibrium point on the U, = U,((,) curve132 MECHANICS OF DISPERSIONSbuild-up would take different paths depending on whether the dispersion is at rest,under flow or (small amplitude) vibration.This observation could be utilized inchemical engineering design for it is important to note that if energy conservation is adominant factor in the handling of thixotropic systems, equilibrium conditions areto be avoided.J4FIG. 2.-Energy (thermal plus shear field) as a function of extent of reaction. Curve 1, U, = Ue(te)in the t = 00 plane, the plane of the paper. Projected on the f = 00 plane, curves 2, thixotropicbuild-up under flow ; 3, break-down, under flow ; 4, build-up, under low amplitude vibration ; and5, build-up at rest. The t-axis is perpendicular to the plane of the paper.There are two limiting equilibrium cases for which 5 can be calculated.In one,the continuum phase is totally disregarded, when the particulate solid becomes agaseous assembly of structureless identical particles, each of mass m. There is now aunique value of 5 for all rates of strain, and the quantity C Q(5) exp( - U,/RT) is thepartition fuizction of the assembly, which has the value V(27-cmNkT/h2)3, where ti isPlanck's constant.18 We can use the expression,rto show that the stress is independent of the rate of strain and hence to derive the well-known result that the coefficient of viscosity of the assembly is independent ~f stress(which in this relation is equal to the pressure, p , of the gas). Such behaviour isfound in near perfect gases.lg In the other case, again disregarding the liquidcontinuum, we cause the particles to gel to an assembly of N identical lacalizedoscillators regularly spaced on a lattice.The value of 5 is again unique for sr!ch anassembly, and at high temperature the partition function for an oscillator is kT/Av,where v is the natural frequency.20 The response to a siiiusoidal strain i s itselfsinusoidal, falling rapidly to zero when the imposcd frequency is much abovc v.This kind of response is found with crossed-linked polymers and gels,14 as s z m intheir relaxation spectra.Implicit in the statistical method we have used is that 5 is a necessary parameter.It follows, therefore, that is a necessary parameter in any study of the rlheology ofthixotropic dispersions.Since the change in the character of the spectra is employeA. J. B. SPAULL 133to follow the way the model changes with time, we have the iniportant result that theanalysis is not dependent on the model. The only constraint on the model is that itmust be thermodynamically reversible (between true and false or between differentfalse equilibrium states) with respect to U, or the applied stress, in time, whichemphasises, in this light, the physical nature of 5, a relaxation parameter.To conclude, we briefly consider a few illustrations where the basic theory mightbe further exploited. It should be possible to calculate the value of 5 and its timedependence for some systems ; armed with such information we might be in a positionto derive a general theory for thixotropic kinetics, and the seat of elasticity: inparticular it would be interesting to use the theory to contrast the effects stemmingfrom the explanation we give for structural hysteresis with those originating fromthe explanation Everett proposed for the hysteresis found in surface chemistry.211 am indebted to Prof.J. Mewis for hospitality in Belgium, during tenure of aRoyal Society European Science Exchange Programme visiting fellowship.APPENDIXFlow time, zf, is the time basis for defining rate of strain tensor, &(= de/dz,), andrate of stress, fiij(= dpij/dzf).Kinematic or kinetic time, K , is the time used in practical experiments, to measurethixotropic change, during build-up or break-down.Tlzermodynamic (or de Donder) time, t, is that used in thermodynamic discussion.The parameter 5 is a function of time, t, only, 5 = t(t).We find the form of thefunction. For the geometry employed in this paper, the arbitrary law U = U(z,)governing the way the energy of the assembly increases is determined from the rate ofstrain tensor, 6 = l/zf.5 = v ( ~ , t) = v(t, 5) = alt,dtWe need to distinguish between different kinds of tinie in rheology.Thus we know the form of the differential equation :where a is a dimensionless constant [see ref. (7), chap. 1, article 91. We adopt twoboundary conditions. One, the rheological ground and thermodynamic standardequilibrium states, cE*s, is the value of 5 when t = 1, and after the dispersionhas been at rest for infinite time, t.We equate t f 9 S arbitrarily to zero. The otherboundary is that for which d</dt = 0. At this boundary t = c,, a false thermo-dynamic equilibrium. Therefore, < = aln(t). The reference state, t:,', is a truethermodynamic equilibrium [see ref. (7), p. 401, since dtE,s/dt = 0, and the affinity,A:*s, is zero. Other equilibrium values of t are false thermodynamic equilibria,since for these values of 5, though dt,/dt = 0, the affinity has a finite positive value.Thus to each value of 5, in the equilibrium curve U, = Ue(te) corresponds a value ofthe affinity, A,. It should be noted that the values of <, in the t = plane can bereflected in the t = 1 plane, fig. 2. Further we have assumed the convention thatdc/dt is positive in sign when a thixotropic path is from a lower to a higher value ofU and negative in a reverse direction.Arbitrarily, the values of 5 are put equal tozero in the t = 1 plane and to 1 in the t = 01) plane.J. Mewis, A. J. B. Spaull and J. Heken, Nature, 1975, 253, 618.D. C-13. Cheng, J. P/zys. D, 1974, 7, L155.J. Mewis and A. J. B. Spaull, Adv. Colloid. Interface Sci., 1976, 6, 173.D. H. Everett, Pure Appl. Chem., 1972, 31, 579.E. A. Guggenheim, Tlternzodynamics (North-Holland, Amsterdam, 5th revised edn, 1967),chap. 2134 MECHANICS OF DISPERSIONSW. H. Bauer and E. A. Collins, Rheology, Theory and applications, ed. F. R. Eirich (AcademicPress, N.Y., 1967), 4, 423.L. Dintenfass, Proc. 5th Int. Congress of Rheology, ed. S . Onogi (University of Tokyo Press,Tokyo, 1970), vol. 2, p. 281.H. Freundlich, Thixotropy (Hermann, Paris, 1935).Ukr. SSSR, 1972).1973), vol. 1, chap. 5.R. Defay, translated by D. H. Everett (Longman, London, 1973), p. 10.' M. Reiner and G. W. Scott-Blair, see ref. (6), p. 461.lo M. N. Kruglitshii and N. V. Mihailov, Rheology of thixotropic systems (Nauk. Dumka., Kiev,l1 R. H. Ottewill, CoZloid science (Specialist Periodical Reports, The Chemical Society, London,l2 For a bibliography of the work of de Donder see Chemical thermodynamics, I. Prigogine andl3 Personal communication from J. Mewis.l4 J. D. Ferry, Viscoelastic properties of polymers (Wiley, New York, 2nd edn, 1970) ; FarahyDisc. Chem. Soc., 1974, 57.G. Schoukens, J. Mewis and A. J. B. Spaull, report presented at a meeting of the British Societyof Rheology, April, 1975.l6 J. Helsen, G. Schoukens, J. Mewis and A. J. B. Spaull, report presented at a meeting of theFaraday Division, Colloid and Interfacial Science Group, Bristol, April, 1976.l7 J. W. Goodwin, Colloid science (Specialist Periodical Reports, The Chemical Society, London,1975), vol. 2, chap. 7.G. S . Rushbrooke, Introduction to statistical mechanics (Oxford U.P., London, 1949), p. 63.l9 J. 0. Hirschfelder, C. F. Curtiss and R. B. Bird, Molecular theory ofgases and Ziquids (Wiley,New York, 1954).2o See ref. (18), p. 65.21 D. H. Everett and P. Nordon, Proc. Roy. Soc. A , 1960, 259, 341.Gels and Gelling Processes.(PAPER 6/477

 

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