Well, isn't this a bit of a tricky question, one that I imagine may have a few valid answers depending upon how the analysis is approached (though I am not suggesting "any interpretation" is as "good as" "any other" interpretation: analysis must be valid and many aren't). Turgenev is a deep thinker with deeply complex text. Many strands are twisted together like the strands of silk embroidery floss: try to pull one strand from a length of floss, with eight to sixteen strands tightly twisted together, and the twist resists separation. Having waxed lyrical, what do we know?
We know that there are fourteen separate events of "unannounced" visits (if sudden death can be considered an unannounced visit, thirteen otherwise) though some are preceded by general invitations or follow previously missed appointment times (the visit to the governor, though unannounced, was common practice).
Turgenev sets up a pattern from the beginning of young men who profess to be nihilists, believers in nothing but scientifically demonstrable externalities of a mechanistic nature ("The machine's out of gear."), who defy social order by arriving where they will when they will but, ironically, not without some adverse consequences, perhaps to illustrate the inexorable laws of mechanistic nature that hold sway over human actions.
The following are the wholly unannounced events though two are mitigated by expectation of some sort.
- Bazarov's initial visit to Nikolai
- Bazarov and Arkady to Madame Odintsov at Nikolskoe
- Bazarov and Arkady to Nikolai after leaving Nikolskoe in disgrace
- Arkady to Katya at Nikolskoe
- Bazrov from Nikolai to Arkady at Nikolskoe
- Bazarov to Parent's again
- Madame Odinsov with doctor to Bazarov
We also know that, regardless of other complexities, the crux of the story is Bazarov's love for Madame Anna Sergeyevna Odintsov and Arkady's eventual love for Katya. We also know that belief in love causes Bazarov to betray his usual nihilistic deportment and try to make himself seem to be more than he is in Madame Odintsov's eyes; this is why Arkady is repeatedly surprised during their stay at Nikolskoe.
again a surprise for Arkady ... [who] was fated to meet with surprises that day.
We also know that Bazarov becomes cynically and bitterly entrenched in his nihilism--adding an emotional, non-mechanistic layer to his belief, a layer exposed on his deathbed when he says his last words to Anna Sergeyevna--after Anna Sergeyevna rejects his offer of love while at Nikolskoe. This is a cynicism that he might have renounced during their conversation in her garden in front of the temple, with Arkady and Katya listening, had either one been even a little yielding. On his deathbed he confesses all this when he says he love her and that she is young and beautiful, renouncing all they had agreed upon in the garden:
'You see,' pursued Anna Sergyevna, 'you and I made a mistake; we are both past our first youth, I especially so; we have seen life, we are tired; [...] I was right yesterday when I told you we were both old people.
This is the essence of what we know. What can we conclude from this? What is Turgenev's purpose in developing this series of unannounced arrivals for our protagonists? Social form, especially in earlier epochs, is what adheres society into a unit and, consequently, is what is very important to a culture. As a nihilist, Bazarov (and his disciple, Arkady) does not believe in the abstraction of social order any more than he believes in the abstraction of emotional truth. Flouting social order is demonstrating his nihilism, his non-belief. Turgenev wrote Fathers and Sons, in part, to express his approval of nihilism and of Bazarov and of the nihilistic rejection of social order and to demonstrate the cohesiveness of a nihilistic philosophy even while being philanthropic (healing the sick) and even in the face of accidental, disappointed, premature death.
'[The] problem for the giant is how to die decently, ... Never mind; I'm not going to turn tail.'