2.2 Two Phase Structures , One Three-dimensional and One Two - dimensional Grain Boundary Phase .sx 2.2.1 Binary alloys .sx Most metallic alloying elements purposely added to improve properties are highly soluble and do not show a marked tendency to segregate on grain boundaries .sx Those of interest here are mainly non-metallic or metalloid elements present as impurities .sx The grain boundary is here treated as a separate phase which , being very thin , is designated two dimensional ( 2D) .sx Solute elements and vacancies are partitioned between the bulk grain , i.e. the 3D phase , and the grain boundary .sx In an alloy consisting of two elements and two bulk phases , such as alpha beta brass , addition of extra solute element would merely change the relative volumes of the two phases without any change in composition of either .sx This cannot happen in the grain - grain boundary case since the volume ( area ) of the latter is fixed .sx Adding more solute , therefore , increases the concentration in both the 3D and the 2D phase .sx In his original analysis of the thermodynamics of the partition McLean supposed the grain boundary to consist of a fixed number of exactly equivalent sites where solute atoms can exchange with solvent atoms with a drop in energy of the system by an amount delta G per mole so exchanged .sx He arrived at a relationship which has since been widely used :sx formula .sx where X refers to the mole fraction of the segregated solute species on the grain boundary and x its mole fraction in the bulk grain , R is the molar gas constant and T the absolute temperature .sx The value of Xo , the concentration of X at which the boundary appears to be saturated , can be obtained for each system by experiment .sx For some systems saturation occurs at a monolayer of atoms .sx For other systems Xo is a fraction of a monolayer .sx For atoms which segregate on the boundary delta G is negative - the total free energy of the system is reduced when solute atoms fall into grain boundary traps and the term exp(- delta G/RT ) is large and positive .sx X/(Xo-X ) is the ratio of solute to solvent atoms on the grain boundary and x/(1-x ) the ratio in the bulk grain .sx The term exp(- delta G/RT ) is therefore the grain boundary enrichment ratio which can take large positive values up to about 10 4 for values of delta G that are found in practice .sx The equation has the same form as the earlier Langmuir equation for adsorption on the free surface of a solid , but the analogy should not be taken too far , since the environment and constraints on atoms at a grain boundary are very different from those of a free surface .sx The latter is an empty half - .sx co-ordination number of atoms on a surface ranges from 1 to about 8 , while for atoms on a grain boundary there is only a small or zero reduction in co-ordination number compared with the bulk grain value and can be in the range 8 to 11 for close-packed metals .sx McLean assumed that the grain boundary was 3 atoms thick and that the substitutional atomic sites were of three types :sx oversize , undersize and average .sx The sites of each type were assumed to be exactly equivalent so that exchange of a solvent atom for a solute atom always involved the same energy change .sx The sites were therefore assumed to be pre-existing and unchanged by the segregation process .sx Since next nearest neighbour interactions are significant when exchanging a metal atom for a non-metal or metalloid having also a large size difference , the McLean model is valid only for low occupancy , i.e. about 1 to 2% .sx For fracture studies much higher levels of occupancy are of interest , usually in the range of 10 to 100% of a monolayer .sx Experimental results even here have been 'fitted' roughly to the McLean equation although the atomic model is no longer tenable because exchanging a layer of metal atoms for non-metal or metalloid at the boundary must severely change its structure .sx Many examples quoted later show a far from linear relationship between bulk concentration and grain boundary occupancy .sx In spite of the range of powerful experimental techniques available , however , it has not been possible to reveal directly the detailed way in which segregated elements are distributed on grain boundaries and how the structures evolve .sx They may be randomly distributed or in small rafts or even in multilayer plates .sx Also , in areas of strong segregation of two elemental species , they may or may not occupy nearest neighbour sites .sx Preferred models of the boundary and for build up of segregated layers begin in each case with a boundary in which the most disturbed sites are confined to a single mono-atomic layer .sx On the centre layer there are only a few sites for substitution of solute atoms where exchange would involve an energy change of delta G max .sx These sites can be arbitrarily chosen to be the large sites which acquire large solute atoms in place of solvent atoms .sx When the solute atom arrives the neighbouring atoms relax outwards , i.e. the two grains are prized part , so that what were the next largest sites become as large as the pre-existing large sites .sx These are now , in turn , occupied by solute atoms , so that the boundary is further prized apart , thus creating new large sites .sx Thus the value of delta G remains constant .sx If 'large' sites are created immediately adjacent only to segregated solute atoms , then the layer will develop by nucleation and growth of platelets .sx Otherwise , for more general grain boundary widening the development might go through a series of ordered arrangements .sx Large atoms might find large sites by dropping into vacant sites on the boundary and diffusing within the boundary .sx It is obviously unsatisfactory to discuss the segregation up to high occupancy entirely in terms of the size of the grain boundary sites , since each newly arriving impurity atom will alter the electron distribution over many interatomic distances .sx For example , in iron the clean boundary has metallic bonding while the boundary with a high sulphur content has weak bonding across the boundary with ionic and/or covalent bonding in the boundary .sx The fact that delta G remains constant therefore signifies only that , as the boundary fills and adopts a series of new structures and types of bonding , the energy drop for each newly arrived atom remains sensibly constant .sx From the previous discussion it is clear that the delta G = const .sx as the grain boundary fills with impurity is not a special case ; but rather is accidental .sx For phosphorous in steel the constancy has been well established as phosphorous substitutes for iron atoms .sx The same model can be used for the cases where delta G increases or decreases with occupancy , with the modification that as each atom arrives it creates sites respectively greater than or not so great as those which were first occupied or offer a more or a less attractive electron configuration .sx Alternatively , as the boundary fills , the solute atoms take up a series of ordered structures , each more stable than the preceding one , i.e. delta G increases , or it may be less stable , i.e. delta G decreasing .sx These changes may not be smooth but are , perhaps , difficult to detect , except in certain cases where saturation is reached and delta G suffers a large step change in value .sx In the exceptional case of tin in pure iron , saturation does not occur until over two monolayers have been formed .sx The same may be true for iron containing antimony .sx There are three important consequences for systems where delta G increases as the concentration of solute in the boundary increases :sx 1 .sx At constant temperature over a certain range the grain boundary equilibrium content of solute is a step function of the concentration in the bulk grains .sx In an extreme case this would be a step function over part of the range as shown in the diagram , Fig. 6 .sx 2 .sx The temperature range between segregation and desegregation is very narrow .sx Again in an extreme case the gap would almost disappear and the grain boundary transition would approach first order kinetics .sx 3 .sx Since the conditions for near complete segregation or desegregation are so critical , the quality of the grain boundary structure is also critical .sx In the critical range , therefore , some grain boundary facets are near saturation while others remain unsegregated .sx This third effect tends to obscure the effects of 1 .sx and 2 .sx since measurements of grain boundary composition are usually taken over many facets and give an average value for heavily and lightly segregated facets .sx The build up of segregation , therefore , is smoothed out , both as a function of temperature and as a function of concentration within the grains , even when very sharp transitions occur on individual facets .sx Figure 6 shows grain boundary concentration versus bulk concentration for selenium and tellurium in pure iron .sx In each case a factor 3 to 4 in bulk concentration is sufficient to cover the range from low grain boundary occupancy to saturation .sx But even here the measurements are the average of many grain facets , some heavily segregated , others almost free of segregate .sx Therefore the possibility of near first order kinetics is obscured .sx A drop in the delta G value could arise simply from the filling of pre-existing sites , starting with the deepest traps and then filling shallower traps .sx Such an explanation has been given for the segregation of sulphur in Ni 3 Al and Ni 3 ( Ti , Al ) alloys .sx The grain boundary occupancy changed with temperature increase more slowly than was consistent with a single value of delta G. Thus at the higher temperature only the deeper traps were filled .sx Over the temperature range 977 to 1045K there was only a factor 2 decrease in boundary sulphur concentration and the maximum concentration was estimated to be only 5 at .sx %. The pre-existing site model could here be accepted .sx For higher concentrations , however , the interaction of solute atoms on the boundary would have to be conceded .sx The ordered structure of the Ni 3 Al type intermetallic compounds does , nevertheless , give rise to lack of adaptability of atomic positions to the development of close-packed grain boundary structures .sx figure&caption .sx 2.2.2 Ternary alloys .sx Still restricting discussion to 'two phase' systems , i.e. a single bulk solid solution phase plus a grain boundary , the first set of examples include those where both solute species would strongly segregate to grain boundaries if present individually .sx When present together they might either :sx 1 .sx show no interaction and each segregate as though the other were not present , or .sx 2 .sx strongly compete for sites , or .sx 3 .sx segregate co-operatively so that each enhances the segregation of the other .sx An example of this type has not been reported .sx In a typical commercial steel there are many elements on grain boundaries in small amounts which may belong to the first type , but the best example is from the classical work of Seah and Hondros who demonstrated in pure iron that tin up to a monolayer and sulphur up to 50 at .sx % coverage each segregated as though the other were not present , i.e. the level of each was not affected by the presence of the other .sx Whether the two elements segregate on different parts of the boundary with tin forming a double layer , whether sulphur is at the tin-iron interface or whether the two elements form a solid solution or compound layer is open to speculation .sx The interstitial elements carbon , nitrogen and boron , so important in ferrous metallurgy , segregate strongly to ferrite grain boundaries .sx Unlike the substitutional segregants their presence in steels has a number of beneficial effects , including their ability to strengthen grain boundaries and so counteract the effect of embrittling species .sx These elements in ferrite exchange interstitial sites in the grains for interstitial sites in the boundary .sx The enrichment ratio can be very high since interstitial sites in the b.c.c. structure are small .sx F.c.c austenite has larger interstitial sites and grain boundary enrichment is consequently less .sx Competition between carbon and phosphorus on the boundaries of ferrite is of great practical importance since carbon strengthens and phosphorus embrittles boundaries .sx Of the two , carbon has the greater affinity for the grain boundary sites and reduces the amount of phosphorus segregated .sx