The relation thermal conductivity is independent surface area but the relation is linear with grain size

For any finite system the phonon spectrum will be discrete (think of the eigen-modes of a string fixed at its two ends, which is restricted to vibrations over a countable set of eigen-frequencies) and this will show up in the thermodynamic properties of the system at low temperatures; here, the larger the size of the grain, the lower the temperatures over which the finite size of the grain will become noticeable in its thermodynamic properties. For a grain of a given finite size, below a certain temperature one will observe a series of activated behaviours (the length of the temperature intervals separating the activation events are of the order of E_g/k_B, where E_g is the relevant gap in the phonon spectrum; think of the eigen-frequencies of the above-mentioned string). In particular, the phonon specific heat will show exponential behaviour over some ranges of temperature T (explicitly, it is almost constant over these intervals and then quickly rises to higher values at certain temperatures), instead of a steady power-low increase (recall that in three-dimensional macroscopic systems, the phonon specific heat steadily increases like T^3). This behaviour will directly influence thermal conductivity, since the phonon contribution to thermal conductivity is proportional to the phonon specific heat. To summarise, for a given grain of a finite size, beginning at very low temperatures, one will observe a thermal conductivity that as function of temperature will initially look like a staircase (sudden jump, then almost constant, then another sudden jump, etc.).

Above I have neglected the electronic contribution to thermal conductivity. For finite systems, the spectrum of electronic excitations is also discrete. If the energy gap separating the ground state from the excited states is larger than the above-mentioned gaps in the phonon frequencies, the above description remains valid. If not, one will have a mixture of two contributions, electronic and vibrational. Recall that the electronic specific heat of metals generally (not always) increases like T for increasing T at low temperatures, so that in metals at very low temperatures the electronic contribution to thermal conductivity always dominates the phonon contribution.

Incidentally, my above argument is focused on a single grain. In practice where one works with many grains, the variation in the sizes of the grains (the dispersity) is likely to wash out the behaviour I have described above (since one measures an ensemble average); for this behaviour to be observable, it is required that all grains be almost identical.

The relation thermal conductivity is independent surface area but the relation is linear with grain size

The relation will depend on the material, temperature... Sometimes grain boundaries (grain size) will play a role, sometimes they will be negligible. For phonon contribution to the thermal conductivity, one generally considers the phonon mean free path, and the different mechanisms of phonon scattering.

http://users.mrl.illinois.edu/cahill/mrs_sp12.pdf

http://www-sp.phy.cam.ac.uk/~wa14/camonly/statistical/Lecture13.pdf

Thermal conductivity has two parts: 1) electronic part of thermal conductivity is related to electrons and holes transporting heat; 2) lattice part of thermal conductivity which is related to the photons travelling through the lattice. The smaller grain size, the more phonon scattering; therefore, thermal conductivity will decrease.

Thermal conductivity is not physically related to the surface area, however, for measuring thermal conductivity, surface preparation (porosity, roughness and etc.) can affect on preciseness of the data.

Yes thermal conductivity increases with increasing grain size

Thank you very much for the answers! :)

i have another doubt on this..! Thermal conductivity is also related to structure right ?.. so how it is related when comes to structure and grain size at the same time?

Crystal structure is an inherent property of a material and you cannot change it. However grain size or crystallite size is changeable and it depends on production (synthesis) process. Any kind of non-uniformity (defect) can restrict the phonon movement. Grain boundaries are considered as defect because they disarrange the crystal uniformity on a grain. Decreasing the grain size means that you are increasing the grain boundaries, so you will decrease the thermal conductivity.

yes it is related to structure. Anything that can scatter phonons or electrons will affect thermal conductivity like impurities, disorder, vacancies, phonons. Normally one considers the relaxation time or mean free path and sum the different contributions (1/t=1/t1+1/t2+...). If some scattering processes have a decay time much higher than another process, they can be neglected. Sometimes scattering at grain boundaries will be a significant contributing process, sometimes not. And for the same material it can happen that at one temperature it will be significant and at another one it will not. If you can get this book, I will recommend you to have a look at it.

http://books.google.ch/books?id=I9XRU9mtBeoC

The relation between thermal conductivity and grain size of a material is linear. Because, when you decrease the grain size of a material the number of grain boundaries are increases. The phonons are scattered at the grain boundaries (GBs) due to misalignment of lattice at the GBs.

So, the thermal conductivity of a material decreases with increasing the GBs and this is one of the facts for low thermal conductivity in the nano structured material.

Thermal conductivity has no relation with surface area of a material.

large phonon scattering in nanomaterials , i.e.low mean free path and low thermal conductivity