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**Phisics and Mathematics**

S = Sadd + dS = Si=1n aiSi + dS, (Si=1n ai = 1),

where the symbols ai and Si are denotes the volume share and individual property of i-th CM component and the Sadd is the property of CM, which may be calculated by certain additive scheme. Then the symbol dS is the value of synergic effect of property S in CM under friction and wear.

All components of the CM with anti-frictional property and firmness for wear are corresponds of their functional fixing, namely: the solid (sol), the binding (bin) and the lubricant (lub) component. With this functional division in mind the formula (1) may be presented as

S = Ssol – abin (Ssol – Sbin) – alub(Ssol – Slub) + dS, (asol + abin + alub = 1).

The symbols a and S are the main values of the volume shares and the individual property of corresponding CM components.

In accordance with addition model the CM surface is the set of the phase surfaces of solid, binding and lubricant CM components in the following correlation asol : abin : alub. If assume that the more plastic lubricant CM component is spreads on the surface of solid and binding components and penetrates into its inter-crystal space, then the surface of the CM under friction and wear is the micro-layer of the clear lubricant component. Taking into account this “concentration wave” is constantly during all friction process by even distribution of components into CM volume (gradasol = gradabin = gradalub = 0). Then the effective concentration of the lubricant component is

alub, eff = alub + Da,

where the Da = Dx grad(wb) = w Dx gradb.

The value w = A alub (1 – alub) is the probability of the realization of “concentration wave” effect [1-5]. The rate setting of probability by condition (w|alub = 1/2)max = 1 is the main way of the constant A fixing: A = 4. If the value gradb is constant then

Dx gradb = Db = (1 – aсм) – k (1 – aсм) = (1 – aсм)(1 – k).

The parameter k is the dimensional factor and the main characteristic which determines the relationship between particle size of solid CM component rsol and the width of the “concentration wave” Dx, i.e. k = rsol /(Dx + rsol).

Finally, we have Da = 4 (1 – k) alub(1 – alub)2 and the value of synergic effect is following

dS = S – Sadd = [(Ssol – Slub) – abin (Ssol – Sbin )/(1 – alub)] Da.

The relative value of synergic effect is the next relation

dS/(Ssol – Slub) = [1 – abin (Ssol – Sbin )/(1 – alub) (Ssol – Slub)] Da.

If the binding CM component is the part of solid CM component or it’s absent (abin = 0) then the relative value of synergic effect may be presented as dS/(Ssol – Slub) = Da.

The value Da may be determined because the concentration parameter alub and the dimensional factor k are knows.

Obviously, the value Dx is the characteristic of the micro-layer “width” of lubricant CM component. It’s necessary that the micro-particles (all or partially) of solid CM component must to be nano-structural fragments for execution of the rтв @ Dx condition. Some of these fragments with sphere or cylindrical forms are the share of solid CM component, which may be taken into consideration as the supplementary lubricant CM component. If the volume share of these possible nanofragments is kn then we have

alub, eff = alub + Da = alub + 4 alub(1 – alub) Db,

Db = (1 – alub) – k (1 – a lub) (1 – kn) = (1 – a lub)(1 – k(1 – kn).

Obviously, the dimension factor k is 0,5, because the rтв @ Dx condition is observed. Therefore, we have Db = 0,5 (1 – aсм) (1 + kн).

Then the relative value of synergic effect is the following relation

dS/(Sтв – Sсм) = Da = 2 aсм (1 – aсм)2 (1 + kн).

With these results in mind the value Da = 4 aсм(1 – aсм)2 may be obtained by kn = 1.

The quantity of the Da in formula (10) is depends from nanostructural parameter kn. With change of kn from 0 to 1 the composition dependences Da(a) are founds the intensification of integral intensively of the synergic effect and the value (Da|a=2/3)max is increase from 0,296 to 0,592.

The changes of the optimal concentration asol, opt º aopt quantity (which provides the maximum positive effect Da, i.e. the function [Da – (1 – a)]) from parameters k and kn were obtained. In partially, with change of dimension factor k from 0,7 to 0,5 the quantity of the (1 – aopt) is increase from 0,05 to 0,12. And by increase nanostructural parameter kn from 0 to 0,6 the value (1 – aopt) is change from 0,12 to 0,21.

The main properties of the firmness for wear and anti-frictional CM are the velocity of mass wear I = (m/x) Ilin and friction coefficient f. The values mCM and xCM denotes the mass and the width of the wearer CM layer, respectively, and the value Ilin is the velocity of line wear, which may be presented by following relation

Ilin = a sol Ilin, sol + (1 – a sol) Ilin, lub + Db (Ilin, sol – Ilin, lub).

Respectively, the friction coefficient is

f = asol fsol + (1 – asol) flub – Db (fsol – flub).

Then the value Db is presented by the next relation Db = a sol (1 – k (1 – kn)).

Thus, both modifications of the “concentration wave” model are the supplements to with each other, because they are describes the micro-particles dimensions of the solid CM component by parameters k and kn variously [4, 5].

2. Ivanov V.V., Balakai V.I., Kurnakova N.Yu. et al. // Rus. J. Appl. Chem., 2008. Т.81. № 12. С.2169-2171.

3. Balakai V.I., Ivanov V.V., Balakai I.V., Arzumanova A.V. // Rus. J. Appl. Chem., 2009. Т.82. №.5. С.851-856.

4. Ivanov V.V. // International journal of experimental education, 2014. - №4.- Part 2. – С.58-59.

5. Ivanov V.V. // International journal of experimental education, 2014. - №4.- Part 2. – С.59-60.

Ivanov V.V. SYNERGISM EFFECT IN COMPOSITIONAL MATERIALS AND COATINGS UNDER FRICTION AND WEAR . International Journal Of Applied And Fundamental Research. – 2014. – № 2 –

URL: www.science-sd.com/457-24683 (16.12.2019).